Calculus Examples

$y = \frac{(4 x + 2)}{5 x}$

Step 1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{4 x + 2}{5 x}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{4 x + 2}{x}]$ .

$\frac{1}{5} \frac{d}{d x} [\frac{4 x + 2}{x}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 x + 2$ and $g (x) = x$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x + 2$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [2]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $4$ by $1$ .

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

Step 3.5

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

$\frac{1}{5} \cdot \frac{x (4 + 0) - (4 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4$ and $0$ .

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

Step 3.6.2

Move $4$ to the left of $x$ .

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

Step 3.7

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

$\frac{1}{5} \cdot \frac{4 x - (4 x + 2) \cdot 1}{x^{2}}$

Step 3.8

Combine fractions.

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Step 3.8.1

Multiply $- 1$ by $1$ .

$\frac{1}{5} \cdot \frac{4 x - (4 x + 2)}{x^{2}}$

Step 3.8.2

Multiply $\frac{1}{5}$ by $\frac{4 x - (4 x + 2)}{x^{2}}$ .

$\frac{4 x - (4 x + 2)}{5 x^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{4 x - (4 x) - 1 \cdot 2}{5 x^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $4$ by $- 1$ .

$\frac{4 x - 4 x - 1 \cdot 2}{5 x^{2}}$

Step 4.2.1.2

Multiply $- 1$ by $2$ .

$\frac{4 x - 4 x - 2}{5 x^{2}}$

Step 4.2.2

Subtract $4 x$ from $4 x$ .

$\frac{0 - 2}{5 x^{2}}$

Step 4.2.3

Subtract $2$ from $0$ .

$\frac{- 2}{5 x^{2}}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{2}{5 x^{2}}$