Calculus Examples

$y = 5 \cos^{- 4} (x)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos^{- 4} (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos^{- 4} (x)]$ .

$5 \frac{d}{d x} [\cos^{- 4} (x)]$

Step 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 $g (x) = \cos (x)$ .

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Step 2.1

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

$5 (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [\cos (x)])$

Step 2.2

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

$5 (- 4 u^{- 5} \frac{d}{d x} [\cos (x)])$

Step 2.3

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

$5 (- 4 \cos^{- 5} (x) \frac{d}{d x} [\cos (x)])$

Step 3

Multiply $- 4$ by $5$ .

$- 20 (\cos^{- 5} (x) \frac{d}{d x} [\cos (x)])$

Step 4

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

$- 20 \cos^{- 5} (x) (- \sin (x))$

Step 5

Multiply $- 1$ by $- 20$ .

$20 \cos^{- 5} (x) \sin (x)$

Step 6

Simplify.

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Step 6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$20 \frac{1}{\cos^{5} (x)} \sin (x)$

Step 6.2

Combine $20$ and $\frac{1}{\cos^{5} (x)}$ .

$\frac{20}{\cos^{5} (x)} \sin (x)$

Step 6.3

Multiply by $1$ .

$\frac{20}{\cos^{5} (x) \cdot 1} \sin (x)$

Step 6.4

Separate fractions.

$\frac{20}{1} \cdot \frac{1}{\cos^{5} (x)} \sin (x)$

Step 6.5

Convert from $\frac{1}{\cos^{5} (x)}$ to $\sec^{5} (x)$ .

$\frac{20}{1} \sec^{5} (x) \sin (x)$

Step 6.6

Divide $20$ by $1$ .

$20 \sec^{5} (x) \sin (x)$