Calculus Examples

$y = a^{3} + \cos^{3} (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $a^{3} + \cos^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [a^{3}] + \frac{d}{d x} [\cos^{3} (x)]$ .

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

Step 1.2

Since $a^{3}$ is constant with respect to $x$ , the derivative of $a^{3}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\cos^{3} (x)]$

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

$0 + \frac{d}{d u} [u^{3}] \frac{d}{d x} [\cos (x)]$

Step 2.1.2

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

$0 + 3 u^{2} \frac{d}{d x} [\cos (x)]$

Step 2.1.3

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

$0 + 3 \cos^{2} (x) \frac{d}{d x} [\cos (x)]$

Step 2.2

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

$0 + 3 \cos^{2} (x) (- \sin (x))$

Step 2.3

Multiply $- 1$ by $3$ .

$0 - 3 \cos^{2} (x) \sin (x)$

Step 3

Subtract $3 \cos^{2} (x) \sin (x)$ from $0$ .

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