Calculus Examples

$\frac{6 x}{x + 5}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{x + 5}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{x + 5}]$ .

$6 \frac{d}{d x} [\frac{x}{x + 5}]$

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 $g (x) = x + 5$ .

$6 \frac{(x + 5) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

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 $n = 1$ .

$6 \frac{(x + 5) \cdot 1 - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

Step 3.2

Multiply $x + 5$ by $1$ .

$6 \frac{x + 5 - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

Step 3.3

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

$6 \frac{x + 5 - x (\frac{d}{d x} [x] + \frac{d}{d x} [5])}{{(x + 5)}^{2}}$

Step 3.4

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

$6 \frac{x + 5 - x (1 + \frac{d}{d x} [5])}{{(x + 5)}^{2}}$

Step 3.5

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

$6 \frac{x + 5 - x (1 + 0)}{{(x + 5)}^{2}}$

Step 3.6

Simplify terms.

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Step 3.6.1

Add $1$ and $0$ .

$6 \frac{x + 5 - x \cdot 1}{{(x + 5)}^{2}}$

Step 3.6.2

Multiply $- 1$ by $1$ .

$6 \frac{x + 5 - x}{{(x + 5)}^{2}}$

Step 3.6.3

Subtract $x$ from $x$ .

$6 \frac{0 + 5}{{(x + 5)}^{2}}$

Step 3.6.4

Add $0$ and $5$ .

$6 \frac{5}{{(x + 5)}^{2}}$

Step 3.6.5

Combine $6$ and $\frac{5}{{(x + 5)}^{2}}$ .

$\frac{6 \cdot 5}{{(x + 5)}^{2}}$

Step 3.6.6

Multiply $6$ by $5$ .

$\frac{30}{{(x + 5)}^{2}}$