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Practice: Chain Rule Find the derivative of y=ln(arctan x)

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Inverse Trig Functions

3 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

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Practice: Chain Rule

Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
More Derivatives of Inverse Trig Functions Questions:
Inverse trigonometric derivative
Find the derivative of g(x)=(cos⁡−1x)(sin⁡−1x)g(x)=\left(\cos^{-1}x\right)\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Given that f(x)=2−arcsin⁡(x)f\left(x\right)=2-\arcsin\left(x\right), find f′(0)f'\left(0\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin⁡(x+2)f\left(x\right)=\arcsin\left(x+2\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right)
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)'
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
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
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
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
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
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Derivatives: Inverse Trigonometric Functions
Find f′(x)f'(x) if f(x)=arctan⁡xf(x)=\arctan \sqrt{x}
Differentiate the following
y=tan⁡−1x2−1y=\tan^{-1}\sqrt{x^2-1}
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)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
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).
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Find the derivative of f(x)=(arccos⁡x)2sec⁡xf\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}.
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
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)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Linear Approximation and Trigonometric Derivative
a) Show that for f(x)=arcsin⁡(x)f(x) = \arcsin(x) , that f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}} .
b) Use linear approximation at a suitable close value to estimate arcsin⁡(0.1)\arcsin(0.1) . Your solution may be left in terms of fractions.
Derivatives: Inverse Trigonometric Functions
Find the derivative of the following function.
f(x)=1arcsin(x)+arcsin(1x)\displaystyle f(x)=\frac{1}{\text{arcsin}(x)}+\text{arcsin}\left(\frac{1}{x}\right)
Inverse Trigonometric Functions
Calculate the derivative of the following functions.
f(x)=arcsin(ex+1)\displaystyle f(x)=\text{arcsin}(e^{x+1})
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}
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.
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
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
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)
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}))