Calculus Examples

$y = \cos (a^{8} + x^{8})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $a^{8} + x^{8}$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [a^{8} + x^{8}]$

Step 1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- \sin (u) \frac{d}{d x} [a^{8} + x^{8}]$

Step 1.3

Replace all occurrences of $u$ with $a^{8} + x^{8}$ .

$- \sin (a^{8} + x^{8}) \frac{d}{d x} [a^{8} + x^{8}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $a^{8} + x^{8}$ with respect to $x$ is $\frac{d}{d x} [a^{8}] + \frac{d}{d x} [x^{8}]$ .

$- \sin (a^{8} + x^{8}) (\frac{d}{d x} [a^{8}] + \frac{d}{d x} [x^{8}])$

Step 2.2

Since $a^{8}$ is constant with respect to $x$ , the derivative of $a^{8}$ with respect to $x$ is $0$ .

$- \sin (a^{8} + x^{8}) (0 + \frac{d}{d x} [x^{8}])$

Step 2.3

Add $0$ and $\frac{d}{d x} [x^{8}]$ .

$- \sin (a^{8} + x^{8}) \frac{d}{d x} [x^{8}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$- \sin (a^{8} + x^{8}) (8 x^{7})$

Step 2.5

Simplify the expression.

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Step 2.5.1

Multiply $8$ by $- 1$ .

$- 8 \sin (a^{8} + x^{8}) x^{7}$

Step 2.5.2

Reorder the factors of $- 8 \sin (a^{8} + x^{8}) x^{7}$ .

$- 8 x^{7} \sin (a^{8} + x^{8})$