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Wize AP Calculus (AB) Textbook > Differentiation 2: Composite, Implicit, & Inverse Functions

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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Example: Trig Derivatives

Find the derivative of the following function

y=sin⁡xy=\sin\sqrt{x}

The outside function is f(x)=sin⁡(x)f(x)=\sin (x) and the inside function is g(x)=x=x−12g(x)=\sqrt{x}=x^{-\frac{1}{2}}.

The chain rule tells us that the derivative is the derivative of the outside function (keeping the inside the same), times the derivative of the inside function:
y′=f′(g(x))×g′(x)y'=f'\left(g\left(x\right)\right)\times g'\left(x\right)
y′=cos⁡(x)×12x−12\displaystyle y'=\cos\left(\sqrt{x}\right)\times\frac{1}{2}x^{-\frac{1}{2}}

Simplifying:
y′=cos⁡(x)2x\displaystyle y'=\frac{\cos\left(\sqrt{x}\right)}{2\sqrt{x}}
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Example: Exponential Derivatives


Find dydx\displaystyle\frac{dy}{dx} given y=2x2\displaystyle y=2^{x^{2}}.

y′=2x2ln⁡2⋅2x y'=2^{x^{2}}\ln2\cdot 2x
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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

Find the derivative of the following function

y=esin⁡xy=e^{\sin\sqrt{x}}