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The chain rule: Logarithmic Derivatives

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Exponential Functions

4 Activities

Wize University Calculus 1 Textbook > Derivatives

Derivatives of Logarithmic Functions

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Wize University Calculus 1 Textbook > Derivatives

The Chain Rule

3 Activities

The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
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}
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: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\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 Logarithmic Functions Questions:
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
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.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
Derivatives: Logarithmic Functions
Calculate the derivative of the following functions.
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Calculate the derivative of the following
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Derivatives: Logarithmic Functions
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x)=\log_2(x^2+2xe^x)
Differentiate the following:
y=ex2+ln⁡xx\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
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.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
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].
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x) = \log_2(x^2 + 2xe^x).
Derivatives of Logarithmic Functions
Evaluate ddx(4xlog⁡700x)\frac{\text{d}}{\text{d}x}\left(4^x\log_{700}x\right). It is not necessary to simplify your answer.
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Given that f(x)=ecos⁡(ln⁡x)f\left(x\right)=e^{\cos\left(\ln x\right)}, find f′(1)f'\left(1\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Derivatives: Logarithmic Functions
Find all the points for which the tangent line of the function
f(x)=4ln⁡(x)+x2−3x+5f(x)= 4 \ln (x) + x^2 - 3x + 5
has slope equal to 3.
Derivatives of Logarithmic Functions: Normal Lines
Find the equation of the normal line to f(x)=ln⁡x at the point (e,1)f(x)=\ln{x} \text{ at the point } (e,1)
Note: This questions requires knowledge of derivatives of logarithms
Derivatives: Logarithmic Functions
Find the equation of the tangent line to the given point for the following function.
f(x)=ln⁡(x+3x+1)at x=1f(x)=\ln\left(\frac{x+3}{x+1}\right) \text{at}\ x=1
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}}
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
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)
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
(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}))