Calculus Examples

$y = \frac{x}{1 - x}$

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 $g (x) = 1 - x$ .

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

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 $n = 1$ .

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

Step 2.2

Multiply $1 - x$ by $1$ .

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

Step 2.3

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

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

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.7.2

Multiply $x$ by $1$ .

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

Step 2.8

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

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

Step 2.9

Simplify by adding terms.

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Step 2.9.1

Multiply $x$ by $1$ .

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

Step 2.9.2

Add $- x$ and $x$ .

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

Step 2.9.3

Simplify the expression.

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Step 2.9.3.1

Add $1$ and $0$ .

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

Step 2.9.3.2

Reorder terms.

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