Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 \cos^{- 4} (x) + \sin (x) \cos (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $5 \cos^{- 4} (x) + \sin (x) \cos (x)$ with respect to $x$ is $\frac{d}{d x} [5 \cos^{- 4} (x)] + \frac{d}{d x} [\sin (x) \cos (x)]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- 1$ by $- 4$ .

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

Step 3.2.5

Multiply $4$ by $5$ .

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

Step 3.3

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

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Step 3.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.5

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.7

Add $1$ and $1$ .

$20 \cos^{- 5} (x) \sin (x) - \sin^{2} (x) + \cos (x) \cos (x)$

Step 3.3.8

Raise $\cos (x)$ to the power of $1$ .

$20 \cos^{- 5} (x) \sin (x) - \sin^{2} (x) + \cos^{1} (x) \cos (x)$

Step 3.3.9

Raise $\cos (x)$ to the power of $1$ .

$20 \cos^{- 5} (x) \sin (x) - \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x)$

Step 3.3.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$20 \cos^{- 5} (x) \sin (x) - \sin^{2} (x) + {\cos (x)}^{1 + 1}$

Step 3.3.11

Add $1$ and $1$ .

$20 \cos^{- 5} (x) \sin (x) - \sin^{2} (x) + \cos^{2} (x)$

Step 3.4

Simplify.

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Step 3.4.1

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

$20 \cos^{- 5} (x) \sin (x) + \cos^{2} (x) - \sin^{2} (x)$

Step 3.4.2

Apply the cosine double-angle identity.

$20 \cos^{- 5} (x) \sin (x) + \cos (2 x)$

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

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

$20 \frac{1}{\cos^{5} (x)} \sin (x) + \cos (2 x)$

Step 3.4.3.2

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

$\frac{20}{\cos^{5} (x)} \sin (x) + \cos (2 x)$

Step 3.4.3.3

Multiply by $1$ .

$\frac{20}{\cos^{5} (x) \cdot 1} \sin (x) + \cos (2 x)$

Step 3.4.3.4

Separate fractions.

$\frac{20}{1} \cdot \frac{1}{\cos^{5} (x)} \sin (x) + \cos (2 x)$

Step 3.4.3.5

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

$\frac{20}{1} \sec^{5} (x) \sin (x) + \cos (2 x)$

Step 3.4.3.6

Divide $20$ by $1$ .

$20 \sec^{5} (x) \sin (x) + \cos (2 x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 20 \sec^{5} (x) \sin (x) + \cos (2 x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 20 \sec^{5} (x) \sin (x) + \cos (2 x)$