Calculus Examples

$y = {(1 - x^{- 1})}^{- 1}$

Step 1

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 $g (x) = 1 - x^{- 1}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $1 - x^{- 1}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [- x^{- 1}]$ .

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

Step 2.4

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

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

Step 2.5

Multiply.

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2

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

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

Step 2.6

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

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

Step 3

Simplify.

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Step 3.1

Reorder the factors of ${(1 - x^{- 1})}^{- 2} (- x^{- 2})$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

Write $1$ as a fraction with a common denominator.

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

Step 3.5.2

Combine the numerators over the common denominator.

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

Step 3.5.3

Apply the product rule to $\frac{x - 1}{x}$ .

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

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.7

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

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

Step 3.8

Cancel the common factor of $x^{2}$ .

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Step 3.8.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 3.8.2

Cancel the common factor.

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

Step 3.8.3

Rewrite the expression.

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