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The Chain Rule: Finding a derivative

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Wize University Calculus 1 Textbook > Derivatives

The Chain Rule

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Exponential Functions

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Trig Functions

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Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}

More The Chain Rule Questions:
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
The Chain Rule
Find g′(0)g'(0) for g(x)=f(x2)+f(x),g(x)=\sqrt{f(x^2)} + f(x), given f(0)=2 and f′(0)=1.f(0)=2 \text{ and }f'(0)=1.
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Find g′(0)g^{\prime}(0) for g(x)=f(x2)+f(x)g(x)=\sqrt{f(x^2)}+f(x), given f(0)=2f(0)=2 and f′(0)=1f^{\prime}(0)=1.
The Chain Rule
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
The Chain Rule
Given this table of values for f, g, f′, and g′f,\ g,\ f',\ \text{and}\ g' below, answer the following questions.
xf(x)f′(x)f′′(x)g(x)g′(x)00−1−523π2π10452−2−410π2−3\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Practice: Chain Rule with Given Values
Q:\textbf{Q:} Given that f(1)=5, g(1)=−2, f′(1)=2f(1)=5,\ g(1)=-2,\ f'(1)=2 and g′(1)=−1/2g'(1)=-1/2, find the derivative of f(x)−g2(x)\sqrt{f(x)-g^2(x)} at x=1x=1.
The Quotient and Chain Rules
Let f(x)=x2−6x−3\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}.
The chain rule
Find the derivative of h(x)=log⁡(cos⁡(x))h(x) = \log(\cos(x)). Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
The chain rule
The derivative of y=(2−x7)500y=\left(2-x^7\right)^{500} is
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Find the derivative of h(x)=f(g(x))   if  f(x)=x2+1h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x}.
Find g′(0)g^{\prime}(0) for g(x)=f(x2)+f(x)g(x)=\sqrt{f(x^2)}+f(x), given f(0)=2f(0)=2and f′(0)=1f^{\prime}(0)=1.
Find the derivative of h(x)=f(g(x))   if  f(x)=x2+1h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x}.
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
The Chain Rule
Find g′(0)g'(0) for g(x)=f(x2)+f(x),g(x)=\sqrt{f(x^2)} + f(x), given f(0)=2 and f′(0)=1.f(0)=2 \text{ and }f'(0)=1.
The Chain Rule
Compute the derivative of
f(x)=x4+4x4+4f(x) = \sqrt{x^4 + \frac{4}{x^4}+4}
Practice: Chain Rule
Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Find the derivative of the following function.
f(x)=arcsinx\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Practice: Chain Rule with Given Values
Q:\textbf{Q:} Given that f(1)=5, g(1)=−2, f′(1)=2f(1)=5,\ g(1)=-2,\ f'(1)=2 and g′(1)=−1/2g'(1)=-1/2, find the derivative of f(x)−g2(x)\sqrt{f(x)-g^2(x)} at x=1x=1.
Given the information g(x)=1+[f(x)]2  such that f(1)=2,f′(1)=−3g(x)=\sqrt{1+[f(x)]^2} \ \text{ such that } f(1)=2,f'(1)=-3. Find g′(1).g'(1). Express your answer as a fraction in lowest terms. If the answer is negative, put the negative sign in front of the entire fraction.
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function f(x)=tan⁡(x3)f\left(x\right)=\sqrt{\tan\left(x^3\right)}
The Chain Rule
If g(t)=9+f(t)3g\left(t\right)=\sqrt[3]{9+f(t)} , write an expression for the derivative g′(t)g'\left(t\right)
The Chain Rule
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
(Duplicated)
Find f′(1)f'\left(1\right) if f(x)=x2 g(x)f\left(x\right)=\sqrt{x^2\ g\left(x\right)}, g(1)=1g\left(1\right)=1 and g′(1)=2g'\left(1\right)=2.
(Duplicated)
If f(x)=(g(x)−h(x))5f\left(x\right)=\left(g\left(x\right)-h\left(x\right)\right)^5, g(0)=2, h(0)=0, g′(0)=6g\left(0\right)=2,\ h\left(0\right)=0,\ g'\left(0\right)=6, and h′(0)=−4h'\left(0\right)=-4, find f′(0)f'\left(0\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
Given this table of values for f, g, f′, and g′f,\ g,\ f',\ \text{and}\ g' below, answer the following questions.
xf(x)f′(x)f′′(x)g(x)g′(x)00−1−523π2π10452−2−410π2−3\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
More Derivatives of Exponential Functions Questions:
Derivatives/Antiderivatives
What is the derivative of f(x)=excos⁡xf(x)=e^{x}\cos x?
Derivatives/Antiderivatives
What is the derivative of f(x)=e2xcos⁡3xf(x)=e^{2x}\cos3x?
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Exponential Functions: Evaluating a limit using derivatives
Practice: Evaluating a limit using derivatives
Evaluate lim⁡h→02(2+h)−4h\displaystyle\lim_{h\to 0}\frac{2^{\left(2+h\right)}-4}{h} by finding the derivative of a specific function at a certain point.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Given f(x)=ex+32xf(x)=e^x+3^{2x} , find f′(2).f'(2).
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
Derivatives: Exponential and Logarithmic Functions
Find the derivative of the following function.
f(x)=(x2+xln⁡x)exf(x)=(x^2+x\ln{x})e^{\sqrt{x}}
Calculate the derivative of the following
f(x)=etan⁡(x2)\displaystyle f(x)=e^{\tan{(x^2)}}
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Derivatives: Exponential and Trigonometric Functions
Find the derivative of
f(x)=5x6−sec⁡−1(e2x)+cos⁡−1xf(x)=5x^6-\sec^{-1}(e^{2x})+\cos^{-1}x
Derivatives: Exponential Functions
Find d10dx10(−5x2+4x+50x3−x8−3e3x)\frac{d^{10}}{dx^{10}}(-5x^2+4x+50x^3-x^8-3e^{3x})
Derivatives: Exponential Functions
Find the coordinates of a point on the curve f(x)=xexf(x)=xe^x so that the slope of the tangent line to the curve at this point is parallel to line of x=yx=y.
Derivatives of Exponential Functions
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point (1,e2)(1,e^2)
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f(x)=tan⁡(e2x)f(x)=\tan(e^{2x})
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Differentiate the following:
y=ex2+ln⁡xx\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
Differentiate the following:
f(x)=e9−6x(x−5)4e−x2f(x)=\frac{e^{9-6x}}{(x-5)^4e^{-x^2}}
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Find the equation of the tangent line to the given point for the following function.
f(x)=ex2+x, at P(1,e2)f(x)=e^{x^2+\sqrt{x}},\,\text{at}\,P(1,e^2)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Given f(x)=ex+32xf(x)=e^x+3^{2x} , find f′(2).f'(2).
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Given f(x)=53xf(x)=5^{3x}, find 𝑓′(0).
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
Derivatives/Antiderivatives
Compute the second derivative of f(x)=etan⁡xf(x)=e^{\tan x}.
Derivatives: Exponential Functions
At which point(s) is the tangent line to the graph y=ex+e−2xy = e^x + e^{-2x} horizontal?
Derivatives: Exponential Functions
Determine ddt(txetan⁡x)\frac{\text{d}}{\text{d}t}\left(txe^{\tan x}\right), where xx is a constant.
If f(x)=2x−1+ex−1f(x)=2x-1 +e^{x-1} find f−1(2)f^{-1}(2)
Derivatives: Trigonometric and Exponential Functions
Evaluate ddx(exsin⁡x)\displaystyle \frac{\text{d}}{\text{d}x}\left(e^x\sin x\right).
Find the derivative of f(x)=22e−2x+2f(x)=\displaystyle \frac{2}{2e^{-2x}+2}.
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
For the function f(x)=e2xcos⁡xf(x)=e^{2x}\cos\sqrt{x} , find f′(4).f'(4).
Find the equation of the tangent line to the given point for the following function.
f(x)=ex2+x, at P(1,e2)f(x)=e^{x^2+\sqrt{x}},\,\text{at}\,P(1,e^2)
Given that f(x)=ecos⁡(ln⁡x)f\left(x\right)=e^{\cos\left(\ln x\right)}, find f′(1)f'\left(1\right).
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, g(π2)=2g\left(\frac{\pi}{2}\right)=2, g′(π2)=3g'\left(\frac{\pi}{2}\right)=3, h(π2)=1h\left(\frac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\frac{\pi}{2}\right)=2, find f′(π2)f'\left(\frac{\pi}{2}\right).
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Exponential Functions: Evaluating a limit using derivatives
Practice: Evaluating a limit using derivatives
Evaluate lim⁡h→02(2+h)−4h\displaystyle\lim_{h\to 0}\frac{2^{\left(2+h\right)}-4}{h} by finding the derivative of a specific function at a certain point.
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Each question is worth 3 marks.
Derivatives of Exponential Functions
Compute the derivative of
f(x)=csc⁡(x)+sin⁡(x2)e2x+1f(x) = \frac{\csc(x) + \sin(x^2)}{e^{2x} + 1}
You do not need to simplify your answer.
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Derivatives: Exponential Functions
If f(x)f(x) is differentiable everywhere, find AA and BB.
f(x)={xe(x2+1) if x≥1Ax+B       if  x<1f(x)=\left\{ \begin{array}{ll} \displaystyle xe^{(x^2+1)}\quad\quad\,\text{if}\,x\geq 1 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<1\\ \end{array} \right.
Derivatives: Exponential Functions
Find the equation of the tangent line to the given point for the following function.
f(x)=ex2+x, at P(1,e2)f(x)=e^{x^2+\sqrt{x}},\,\text{at}\,P(1,e^2)
Derivatives: Exponential Functions
Find the derivative of the following function.
y=esin⁡x+sin⁡ex\displaystyle y=e^{\sin{x}}+\sin{e^x}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=ex2sec(x)\displaystyle f(x)=e^{x^2\text{sec}(x)}
Derivatives of Exponential Functions
Find the derivative of f(x)=x3exf(x)=x^3e^x
Derivatives: Exponential Functions
Calculate the derivative of the following functions.
f(x)=etan⁡(x2)\displaystyle f(x)=e^{\tan{(x^2)}}
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
More Derivatives of Trig Functions Questions:
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=ex2sec(x)\displaystyle f(x)=e^{x^2\text{sec}(x)}
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Derivatives: Trigonometric Functions
Find the slope of the tangent line to the following function at x=π/6:x=\pi/6:
g(x)=e−12ln⁡(sin⁡x)g(x)=e^{-\frac{1}{2}\ln(\sin x)}
Derivatives: Trigonometric Functions
Given that g(1)=g′(1)=1g(1)=g'(1)=1, find f′(π/2) if f(x)=g(sin⁡x)x+1f'(\pi/2) \text{ if }f(x)=\frac{\sqrt{g(\sin x)}}{x+1}
Derivatives: Trigonometric Functions
Find
d22dx22(−10cos⁡(x))\frac{d^{22}}{dx^{22}}(-10\cos(x))
Derivatives: Trigonometric Functions
Find the equation of the tangent line and the normal line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x)=(x^2-2)\sin x+2x\cos x at x=πx=\pi
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f(x)=tan⁡(e2x)f(x)=\tan(e^{2x})
Derivatives: Trigonometric Functions
Find the equation of the line tangent to the graph of f(x)=x3sin⁡(πx)f(x)=x^3\sin(\pi x) at the point x=2.x=2.
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the equation of the tangent line of the graph y=excos⁡ xy=\frac{e^x}{\cos\ x}at the point where 𝑥 = 0.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function f(x)=tan⁡(x3)f\left(x\right)=\sqrt{\tan\left(x^3\right)}
Derivatives of Trigonometric Functions
Determine the point(s), if any, where the function y=cos⁡2x−sin⁡2xy=\cos^2x-\sin^2x has a horizontal tangent line.
Derivatives: Trigonometric and Logarithmic Functions
Find ddy[(y5+ln⁡y)tan⁡y]\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right].
Derivatives: Trigonometric and Exponential Functions
Evaluate ddx(exsin⁡x)\displaystyle \frac{\text{d}}{\text{d}x}\left(e^x\sin x\right).
Find the derivative of the function f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}.
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, g(π2)=2g\left(\frac{\pi}{2}\right)=2, g′(π2)=3g'\left(\frac{\pi}{2}\right)=3, h(π2)=1h\left(\frac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\frac{\pi}{2}\right)=2, find f′(π2)f'\left(\frac{\pi}{2}\right).
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Practice: Chain Rule
Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Derivatives: Trigonometric Functions
Find f′(π/2)f^{\prime}(\pi/2)if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}, where g(1)=g′(1)=1. g(1)=g^{\prime}(1)=1.
Derivatives: Trigonometric Functions
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))