Calculus Examples

$y = \frac{1}{1 - 2 x}$

Step 1

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

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

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 $g (x) = 1 - 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - 2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - 2 x$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - 2 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x]$ .

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 3.5

Multiply $- 2$ by $- 1$ .

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

Step 3.6

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

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

Step 3.7

Multiply $2$ by $1$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Reorder terms.

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