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Wize University Calculus 1 Textbook > Derivatives

The Chain Rule

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The Chain Rule

This differentiation shortcut lets us take the derivative of composed functions. (functions inside functions)

The Chain Rule

The derivative of a composition of two differentiable functions f(y)f(y) and g(x)g(x) (so that the composition f(g(x))f(g(x))is well defined), is
[f(g(x))]′=f′(g(x))g′(x)\boxed{[f(g(x))]'=f'(g(x))g'(x)}


Wize Tip
A nice way to remember the Chain Rule is: "The derivative of the outside (keep the inside) times the derivative of the inside".

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Example: The Chain Rule

Find the derivative of the function y=(2x−5)10y=(2x-5)^{10}

y′=10(2x−5)10−1×2=20(2x−5)9y' =10(2x-5)^{10-1}\times2 \\ \text{} \\=\boxed{20(2x-5)^9}



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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

Extra Practice
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)
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}))