Calculus Examples

$y = - \frac{3}{5 x + 2}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{5 x + 2}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{5 x + 2}]$ .

$- 3 \frac{d}{d x} [\frac{1}{5 x + 2}]$

Step 1.2

Rewrite $\frac{1}{5 x + 2}$ as ${(5 x + 2)}^{- 1}$ .

$- 3 \frac{d}{d x} [{(5 x + 2)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 5 x + 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 x + 2$ .

$- 3 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [5 x + 2])$

Step 2.2

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

$- 3 (- u^{- 2} \frac{d}{d x} [5 x + 2])$

Step 2.3

Replace all occurrences of $u$ with $5 x + 2$ .

$- 3 (- {(5 x + 2)}^{- 2} \frac{d}{d x} [5 x + 2])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $- 3$ .

$3 ({(5 x + 2)}^{- 2} \frac{d}{d x} [5 x + 2])$

Step 3.2

By the Sum Rule, the derivative of $5 x + 2$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [2]$ .

$3 {(5 x + 2)}^{- 2} (\frac{d}{d x} [5 x] + \frac{d}{d x} [2])$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$3 {(5 x + 2)}^{- 2} (5 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.4

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

$3 {(5 x + 2)}^{- 2} (5 \cdot 1 + \frac{d}{d x} [2])$

Step 3.5

Multiply $5$ by $1$ .

$3 {(5 x + 2)}^{- 2} (5 + \frac{d}{d x} [2])$

Step 3.6

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 {(5 x + 2)}^{- 2} (5 + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $5$ and $0$ .

$3 {(5 x + 2)}^{- 2} \cdot 5$

Step 3.7.2

Multiply $5$ by $3$ .

$15 {(5 x + 2)}^{- 2}$

Step 4

Simplify.

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Step 4.1

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

$15 \frac{1}{{(5 x + 2)}^{2}}$

Step 4.2

Combine $15$ and $\frac{1}{{(5 x + 2)}^{2}}$ .

$\frac{15}{{(5 x + 2)}^{2}}$