Calculus Examples

$y = \frac{1}{{(4 x + 7)}^{3}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{{(4 x + 7)}^{3}}$ as ${({(4 x + 7)}^{3})}^{- 1}$ .

$\frac{d}{d x} [{({(4 x + 7)}^{3})}^{- 1}]$

Step 1.2

Multiply the exponents in ${({(4 x + 7)}^{3})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [{(4 x + 7)}^{3 \cdot - 1}]$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\frac{d}{d x} [{(4 x + 7)}^{- 3}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 4 x + 7$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 x + 7$ .

$\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [4 x + 7]$

Step 2.2

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

$- 3 u^{- 4} \frac{d}{d x} [4 x + 7]$

Step 2.3

Replace all occurrences of $u$ with $4 x + 7$ .

$- 3 {(4 x + 7)}^{- 4} \frac{d}{d x} [4 x + 7]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x + 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [7]$ .

$- 3 {(4 x + 7)}^{- 4} (\frac{d}{d x} [4 x] + \frac{d}{d x} [7])$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$- 3 {(4 x + 7)}^{- 4} (4 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 3.3

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

$- 3 {(4 x + 7)}^{- 4} (4 \cdot 1 + \frac{d}{d x} [7])$

Step 3.4

Multiply $4$ by $1$ .

$- 3 {(4 x + 7)}^{- 4} (4 + \frac{d}{d x} [7])$

Step 3.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- 3 {(4 x + 7)}^{- 4} (4 + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4$ and $0$ .

$- 3 {(4 x + 7)}^{- 4} \cdot 4$

Step 3.6.2

Multiply $4$ by $- 3$ .

$- 12 {(4 x + 7)}^{- 4}$

Step 4

Simplify.

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Step 4.1

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

$- 12 \frac{1}{{(4 x + 7)}^{4}}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 12$ and $\frac{1}{{(4 x + 7)}^{4}}$ .

$\frac{- 12}{{(4 x + 7)}^{4}}$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{12}{{(4 x + 7)}^{4}}$