Calculus Examples

$e^{w^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = e^{w}$ and $g (w) = w^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $w^{2}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d w} [w^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d w} [w^{2}]$

Step 1.3

Replace all occurrences of $u$ with $w^{2}$ .

$e^{w^{2}} \frac{d}{d w} [w^{2}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 2$ .

$e^{w^{2}} (2 w)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $e^{w^{2}} (2 w)$ .

$2 e^{w^{2}} w$

Step 3.2

Reorder factors in $2 e^{w^{2}} w$ .

$2 w e^{w^{2}}$