Calculus Examples

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (2 x)$ .

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

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

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

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

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

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

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 $g (x) = 2 x$ .

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

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

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

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

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

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

Differentiate.

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Multiply $- 1$ by $3$ .

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

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Multiply $2$ by $- 3$ .

$- 6 \cos^{2} (2 x) \sin (2 x) \frac{d}{d x} [x]$

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

$- 6 \cos^{2} (2 x) \sin (2 x) \cdot 1$

Multiply $- 6$ by $1$ .

$- 6 \cos^{2} (2 x) \sin (2 x)$