Calculus Examples

$\cos^{2} (8 x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d Z} [f (g (Z))]$ is $f' (g (Z)) g' (Z)$ where $f (Z) = Z^{2}$ and $g (Z) = \cos (8 x)$ .

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To apply the Chain Rule, set $u$ as $\cos (8 x)$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d Z} [\cos (8 x)]$

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

$2 u \frac{d}{d Z} [\cos (8 x)]$

Replace all occurrences of $u$ with $\cos (8 x)$ .

$2 \cos (8 x) \frac{d}{d Z} [\cos (8 x)]$

Step 2

Since $\cos (8 x)$ is constant with respect to $Z$ , the derivative of $\cos (8 x)$ with respect to $Z$ is $0$ .

$2 \cos (8 x) \cdot 0$

Step 3

Combine terms.

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Multiply $0$ by $2$ .

$0 \cos (8 x)$

Multiply $0$ by $\cos (8 x)$ .

$0$