Chapter 2: Differentiation

Section 2.4: The Chain Rule

Example 2.4.3

$$ Use the Chain rule to obtain the second derivative of $F\left(x\right)\=f\left(g\left(x\right)\right)$. $$ Solution $$ •  Obtain the first derivative by the Chain rule. $F\prime \left(x\right)\=f\prime \left(g\left(x\right)\right) g\prime \left(x\right)$ Obtain the second derivative by the Product rule •  The derivative of the first factor requires the Chain rule. $F″\left(x\right)$ $\=f\prime \left(g\left(x\right)\right) g″\left(x\right)\+g\prime \left(x\right) \left[f″\left(g\left(x\right)\right) g\prime \left(x\right)\right]$ $$ $\=f\prime \left(g\left(x\right)\right) g″\left(x\right)\+f″\left(g\left(x\right)\right){\left(g\prime \left(x\right)\right)}^2$ $$ Don't look at this result as a formula, but rather, as a process.$$ $$

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