Calculus Examples

$y = e^{\sin (x^{2})}$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x^{2})$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\sin (x^{2})]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\sin (x^{2})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (x^{2})$ .

$e^{\sin (x^{2})} \frac{d}{d x} [\sin (x^{2})]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$e^{\sin (x^{2})} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{2}])$

Step 2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$e^{\sin (x^{2})} (\cos (u_{2}) \frac{d}{d x} [x^{2}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$e^{\sin (x^{2})} (\cos (x^{2}) \frac{d}{d x} [x^{2}])$

Step 3

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

$e^{\sin (x^{2})} \cos (x^{2}) (2 x)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $e^{\sin (x^{2})} \cos (x^{2}) (2 x)$ .

$2 e^{\sin (x^{2})} x \cos (x^{2})$

Step 4.2

Reorder factors in $2 e^{\sin (x^{2})} x \cos (x^{2})$ .

$2 x e^{\sin (x^{2})} \cos (x^{2})$