Calculus Examples

$\cos^{4} (x) - \sin^{4} (x)$

Step 1

By the Sum Rule, the derivative of $\cos^{4} (x) - \sin^{4} (x)$ with respect to $x$ is $\frac{d}{d x} [\cos^{4} (x)] + \frac{d}{d x} [- \sin^{4} (x)]$ .

$\frac{d}{d x} [\cos^{4} (x)] + \frac{d}{d x} [- \sin^{4} (x)]$

Step 2

Evaluate $\frac{d}{d x} [\cos^{4} (x)]$ .

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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 $f (x) = x^{4}$ and $g (x) = \cos (x)$ .

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Step 2.1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin^{4} (x)]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$4 {u_{1}}^{3} \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin^{4} (x)]$

Step 2.1.3

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

$4 \cos^{3} (x) \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin^{4} (x)]$

Step 2.2

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

$4 \cos^{3} (x) (- \sin (x)) + \frac{d}{d x} [- \sin^{4} (x)]$

Step 2.3

Multiply $- 1$ by $4$ .

$- 4 \cos^{3} (x) \sin (x) + \frac{d}{d x} [- \sin^{4} (x)]$

Step 3

Evaluate $\frac{d}{d x} [- \sin^{4} (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin^{4} (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin^{4} (x)]$ .

$- 4 \cos^{3} (x) \sin (x) - \frac{d}{d x} [\sin^{4} (x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \sin (x)$ .

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Step 3.2.1

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

$- 4 \cos^{3} (x) \sin (x) - (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [\sin (x)])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$- 4 \cos^{3} (x) \sin (x) - (4 {u_{2}}^{3} \frac{d}{d x} [\sin (x)])$

Step 3.2.3

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

$- 4 \cos^{3} (x) \sin (x) - (4 \sin^{3} (x) \frac{d}{d x} [\sin (x)])$

Step 3.3

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

$- 4 \cos^{3} (x) \sin (x) - (4 \sin^{3} (x) \cos (x))$

Step 3.4

Multiply $4$ by $- 1$ .

$- 4 \cos^{3} (x) \sin (x) - 4 \sin^{3} (x) \cos (x)$

Step 4

Simplify.

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Step 4.1

Factor $4 \cos (x) \sin (x)$ out of $- 4 \cos^{3} (x) \sin (x) - 4 \sin^{3} (x) \cos (x)$ .

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Step 4.1.1

Factor $4 \cos (x) \sin (x)$ out of $- 4 \cos^{3} (x) \sin (x)$ .

$4 \cos (x) \sin (x) (- \cos^{2} (x)) - 4 \sin^{3} (x) \cos (x)$

Step 4.1.2

Factor $4 \cos (x) \sin (x)$ out of $- 4 \sin^{3} (x) \cos (x)$ .

$4 \cos (x) \sin (x) (- \cos^{2} (x)) + 4 \cos (x) \sin (x) (- \sin^{2} (x))$

Step 4.1.3

Factor $4 \cos (x) \sin (x)$ out of $4 \cos (x) \sin (x) (- \cos^{2} (x)) + 4 \cos (x) \sin (x) (- \sin^{2} (x))$ .

$4 \cos (x) \sin (x) (- \cos^{2} (x) - \sin^{2} (x))$

Step 4.2

Factor $- 1$ out of $- \cos^{2} (x)$ .

$4 \cos (x) \sin (x) (- (\cos^{2} (x)) - \sin^{2} (x))$

Step 4.3

Factor $- 1$ out of $- \sin^{2} (x)$ .

$4 \cos (x) \sin (x) (- (\cos^{2} (x)) - (\sin^{2} (x)))$

Step 4.4

Factor $- 1$ out of $- (\cos^{2} (x)) - (\sin^{2} (x))$ .

$4 \cos (x) \sin (x) (- (\cos^{2} (x) + \sin^{2} (x)))$

Step 4.5

Rearrange terms.

$4 \cos (x) \sin (x) (- (\sin^{2} (x) + \cos^{2} (x)))$

Step 4.6

Apply pythagorean identity.

$4 \cos (x) \sin (x) (- 1 \cdot 1)$

Step 4.7

Multiply $- 1$ by $1$ .

$4 \cos (x) \sin (x) \cdot - 1$

Step 4.8

Multiply $- 1$ by $4$ .

$- 4 \cos (x) \sin (x)$