Calculus Examples

$y = \cos (a^{6} + x^{6})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = \cos (a)$ and $g (a) = a^{6} + x^{6}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $a^{6} + x^{6}$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d a} [a^{6} + x^{6}]$

Step 1.2

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

$- \sin (u) \frac{d}{d a} [a^{6} + x^{6}]$

Step 1.3

Replace all occurrences of $u$ with $a^{6} + x^{6}$ .

$- \sin (a^{6} + x^{6}) \frac{d}{d a} [a^{6} + x^{6}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $a^{6} + x^{6}$ with respect to $a$ is $\frac{d}{d a} [a^{6}] + \frac{d}{d a} [x^{6}]$ .

$- \sin (a^{6} + x^{6}) (\frac{d}{d a} [a^{6}] + \frac{d}{d a} [x^{6}])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 6$ .

$- \sin (a^{6} + x^{6}) (6 a^{5} + \frac{d}{d a} [x^{6}])$

Step 2.3

Since $x^{6}$ is constant with respect to $a$ , the derivative of $x^{6}$ with respect to $a$ is $0$ .

$- \sin (a^{6} + x^{6}) (6 a^{5} + 0)$

Step 2.4

Add $6 a^{5}$ and $0$ .

$- \sin (a^{6} + x^{6}) (6 a^{5})$

Step 3

Simplify.

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Step 3.1

Multiply $6$ by $- 1$ .

$- 6 \sin (a^{6} + x^{6}) a^{5}$

Step 3.2

Reorder the factors of $- 6 \sin (a^{6} + x^{6}) a^{5}$ .

$- 6 a^{5} \sin (a^{6} + x^{6})$