Chapter 2: Differentiation

Section 2.4: The Chain Rule

Example 2.4.7

$$ Use the Inverse-Function rule to obtain the derivative of $g\left(x\right)\=\sqrt{x}$ from the derivative of $f\left(x\right)\=x^2$. $$ Solution $$ •  The Power rule gives $g\prime \left(x\right)\=\frac{1}{2\sqrt{x}}$ and $f\prime \left(x\right)\=2 x$.   •  The Inverse-Function rule gives   $g\prime \left(x\right)\=\frac{1}{f\prime \left(g\left(x\right)\right)}\=\frac{1}{\genfrac{}{}{0ex}{}{2 x}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=\sqrt{x}}}\=\frac{1}{2\sqrt{x}}$  $$

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