Calculus Examples

$y = \frac{4 x}{x - 5}$

Step 1

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

$4 \frac{d}{d x} [\frac{x}{x - 5}]$

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 $f (x) = x$ and $g (x) = x - 5$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $x - 5$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 3.6

Simplify terms.

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Step 3.6.1

Add $1$ and $0$ .

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

Step 3.6.2

Multiply $- 1$ by $1$ .

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

Step 3.6.3

Subtract $x$ from $x$ .

$4 \frac{0 - 5}{{(x - 5)}^{2}}$

Step 3.6.4

Simplify the expression.

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Step 3.6.4.1

Subtract $5$ from $0$ .

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

Step 3.6.4.2

Move the negative in front of the fraction.

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

Step 3.6.4.3

Multiply $- 1$ by $4$ .

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

Step 3.6.5

Combine $- 4$ and $\frac{5}{{(x - 5)}^{2}}$ .

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

Step 3.6.6

Simplify the expression.

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Step 3.6.6.1

Multiply $- 4$ by $5$ .

$\frac{- 20}{{(x - 5)}^{2}}$

Step 3.6.6.2

Move the negative in front of the fraction.

$- \frac{20}{{(x - 5)}^{2}}$