Calculus Examples

$\sin (x^{2}) \cos (x^{3})$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x^{2})$ and $g (x) = \cos (x^{3})$ .

$\sin (x^{2}) \frac{d}{d x} [\cos (x^{3})] + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x^{3}$ .

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To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$\sin (x^{2}) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$\sin (x^{2}) (- \sin (u_{1}) \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$\sin (x^{2}) (- \sin (x^{3}) \frac{d}{d x} [x^{3}]) + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

Differentiate using the Power Rule.

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Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\sin (x^{2}) (- \sin (x^{3}) (3 x^{2})) + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

Multiply $3$ by $- 1$ .

$\sin (x^{2}) (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) \frac{d}{d x} [\sin (x^{2})]$

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

$\sin (x^{2}) (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{2}])$

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

$\sin (x^{2}) (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) (\cos (u_{2}) \frac{d}{d x} [x^{2}])$

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

$\sin (x^{2}) (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) (\cos (x^{2}) \frac{d}{d x} [x^{2}])$

Differentiate using the Power Rule.

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

$\sin (x^{2}) (- 3 \sin (x^{3}) x^{2}) + \cos (x^{3}) \cos (x^{2}) (2 x)$

Reorder terms.

$- 3 x^{2} \sin (x^{3}) \sin (x^{2}) + 2 x \cos (x^{3}) \cos (x^{2})$