Calculus Examples

$y = {(1 + \sin (7 x))}^{- 4}$

Step 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 $g (x) = 1 + \sin (7 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $1 + \sin (7 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{- 4}] \frac{d}{d x} [1 + \sin (7 x)]$

Step 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 $n = - 4$ .

$- 4 {u_{1}}^{- 5} \frac{d}{d x} [1 + \sin (7 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $1 + \sin (7 x)$ .

$- 4 {(1 + \sin (7 x))}^{- 5} \frac{d}{d x} [1 + \sin (7 x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 + \sin (7 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (7 x)]$ .

$- 4 {(1 + \sin (7 x))}^{- 5} (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (7 x)])$

Step 2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 4 {(1 + \sin (7 x))}^{- 5} (0 + \frac{d}{d x} [\sin (7 x)])$

Step 2.3

Add $0$ and $\frac{d}{d x} [\sin (7 x)]$ .

$- 4 {(1 + \sin (7 x))}^{- 5} \frac{d}{d x} [\sin (7 x)]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 7 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$- 4 {(1 + \sin (7 x))}^{- 5} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [7 x])$

Step 3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$- 4 {(1 + \sin (7 x))}^{- 5} (\cos (u_{2}) \frac{d}{d x} [7 x])$

Step 3.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$- 4 {(1 + \sin (7 x))}^{- 5} (\cos (7 x) \frac{d}{d x} [7 x])$

Step 4

Differentiate.

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Step 4.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$- 4 {(1 + \sin (7 x))}^{- 5} \cos (7 x) (7 \frac{d}{d x} [x])$

Step 4.2

Multiply $7$ by $- 4$ .

$- 28 {(1 + \sin (7 x))}^{- 5} \cos (7 x) \frac{d}{d x} [x]$

Step 4.3

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

$- 28 {(1 + \sin (7 x))}^{- 5} \cos (7 x) \cdot 1$

Step 4.4

Multiply $- 28$ by $1$ .

$- 28 {(1 + \sin (7 x))}^{- 5} \cos (7 x)$

Step 5

Simplify.

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Step 5.1

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

$- 28 \frac{1}{{(1 + \sin (7 x))}^{5}} \cos (7 x)$

Step 5.2

Combine $- 28$ and $\frac{1}{{(1 + \sin (7 x))}^{5}}$ .

$\frac{- 28}{{(1 + \sin (7 x))}^{5}} \cos (7 x)$

Step 5.3

Move the negative in front of the fraction.

$- \frac{28}{{(1 + \sin (7 x))}^{5}} \cos (7 x)$

Step 5.4

Combine $\cos (7 x)$ and $\frac{28}{{(1 + \sin (7 x))}^{5}}$ .

$- \frac{\cos (7 x) \cdot 28}{{(1 + \sin (7 x))}^{5}}$

Step 5.5

Move $28$ to the left of $\cos (7 x)$ .

$- \frac{28 \cos (7 x)}{{(1 + \sin (7 x))}^{5}}$