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Wize AP Calculus (AB) Textbook > Differentiation 1: Basics & Fundamental Properties

The Product Rule

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The Product Rule

Our next differentiation shortcut lets us take derivatives of the product (multiplication) of two functions.

Product Rule

The derivative of a product y=f(x)g(x)y=f\left(x\right)g\left(x\right), where f(x)f(x) and g(x)g(x) are differentiable functions, is:

[f(x)g(x)]′=f′(x)g(x)+f(x)g′(x)\boxed{\quad \left[f\left(x\right)g\left(x\right)\right]'=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)\quad }

Wize Tip
A nice way to remember the product rule is: "the derivative of the 1st times the 2nd, plus the 2nd times the derivative of the 1st".

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Example: The Product Rule

Find the following derivative

ddx[x2(x+7)]\displaystyle\mydd{x^2(x+7)}{x}[bt]


ddx[x2(x+7)]=ddx[x2](x+7)+(x2)ddx[x+7]=(2x)(x+7)+x2(1)=2x2+14x+x2=3x2+14x\displaystyle\mydd{x^2(x+7)}{x}[bt] = \mydd{x^2}{x}[bt](x+7) + (x^2) \mydd{x+7}{x}[bt] \\ \text{} \\ = (2x)(x+7) + x^2(1)=2x^2+14x+x^2 \\ \text{} \\ =\boxed{3x^2+14x}




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Example: The Product Rule

Find the following derivative using the product rule

ddx(x2+1)x2\displaystyle\frac{d}{dx}\frac{(x^2+1)}{x^2}


ddx(x2+1)x2\displaystyle\frac{d}{dx}\frac{(x^2+1)}{x^2} =ddx(x2+1)x−2=\displaystyle\frac{d}{dx}(x^2+1)x^{-2}
=(x2+1)′×x−2+(x2+1)×(x−2)′\displaystyle=(x^2+1)'\times x^{-2}+(x^2+1)\times (x^{-2})'
=2x×x−2+(x2+1)(−2x−3)\displaystyle=2x\times x^{-2}+(x^2+1)(-2x^{-3})
=2x−2(x2+1)x3=\displaystyle\frac{2}{x}-\frac{2(x^2+1)}{x^3}
Find the derivative of y=xf(x)y=\displaystyle xf(x) at x=1x=1 if f(1)=2f(1)=2 and f′(1)=3f'(1)=3