Calculus Examples

$f (x) = \frac{1}{1 - x}$

Step 1

Rewrite $\frac{1}{1 - x}$ as ${(1 - x)}^{- 1}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 1 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [1 - x]$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

Multiply ${(1 - x)}^{- 2}$ by $1$ .

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

Step 3.6

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

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

Step 3.7

Multiply ${(1 - x)}^{- 2}$ by $1$ .

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

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

Reorder terms.

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