Calculus Examples

$f (x) = \frac{x}{2 - 11 x}$

Step 1

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 $g (x) = 2 - 11 x$ .

$\frac{(2 - 11 x) \frac{d}{d x} [x] - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(2 - 11 x) \cdot 1 - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.2

Multiply $2 - 11 x$ by $1$ .

$\frac{2 - 11 x - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $2 - 11 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x]$ .

$\frac{2 - 11 x - x (\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x])}{{(2 - 11 x)}^{2}}$

Step 2.4

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

$\frac{2 - 11 x - x (0 + \frac{d}{d x} [- 11 x])}{{(2 - 11 x)}^{2}}$

Step 2.5

Add $0$ and $\frac{d}{d x} [- 11 x]$ .

$\frac{2 - 11 x - x \frac{d}{d x} [- 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.6

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$\frac{2 - 11 x - x (- 11 \frac{d}{d x} [x])}{{(2 - 11 x)}^{2}}$

Step 2.7

Multiply $- 11$ by $- 1$ .

$\frac{2 - 11 x + 11 x \frac{d}{d x} [x]}{{(2 - 11 x)}^{2}}$

Step 2.8

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

$\frac{2 - 11 x + 11 x \cdot 1}{{(2 - 11 x)}^{2}}$

Step 2.9

Simplify by adding terms.

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Step 2.9.1

Multiply $11$ by $1$ .

$\frac{2 - 11 x + 11 x}{{(2 - 11 x)}^{2}}$

Step 2.9.2

Add $- 11 x$ and $11 x$ .

$\frac{2 + 0}{{(2 - 11 x)}^{2}}$

Step 2.9.3

Simplify the expression.

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Step 2.9.3.1

Add $2$ and $0$ .

$\frac{2}{{(2 - 11 x)}^{2}}$

Step 2.9.3.2

Reorder terms.

$\frac{2}{{(- 11 x + 2)}^{2}}$