Calculus Examples

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

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

The final answer is $\frac{1}{x + h - 1}$ .

$\frac{1}{x + h - 1}$

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h - 1} - (\frac{1}{x - 1})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{x + h - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1}}{h}$

Step 4.1.2

To write $- \frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{x + h - 1}{x + h - 1}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h - 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{1}{x + h - 1}$ by $\frac{x - 1}{x - 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{1}{x - 1} \cdot \frac{x + h - 1}{x + h - 1}}{h}$

Step 4.1.3.2

Multiply $\frac{1}{x - 1}$ by $\frac{x + h - 1}{x + h - 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{x + h - 1}{(x - 1) (x + h - 1)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x - 1) (x + h - 1)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{x + h - 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1 - (x + h - 1)}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5

Rewrite $\frac{x - 1 - (x + h - 1)}{(x + h - 1) (x - 1)}$ in a factored form.

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Step 4.1.5.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1 - x - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.2

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x - 1 - x - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.3

Subtract $x$ from $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{0 - 1 - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.4

Subtract $1$ from $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- 1 - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.5

Add $- 1$ and $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- h + 0}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.6

Add $- h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- h}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$f' (x) = lim_{h \to 0} \frac{- \frac{h}{(x + h - 1) (x - 1)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} - \frac{h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{h}{(x + h - 1) (x - 1)}$ into the numerator.

$f' (x) = lim_{h \to 0} \frac{- h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- h$ .

$f' (x) = lim_{h \to 0} \frac{h \cdot - 1}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h \cdot - 1}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$f' (x) = lim_{h \to 0} \frac{- 1}{(x + h - 1) (x - 1)}$

Step 4.4

Move the negative in front of the fraction.

$f' (x) = lim_{h \to 0} - \frac{1}{(x + h - 1) (x - 1)}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $- 1$ outside of the limit because it is constant with respect to $h$ .

$- lim_{h \to 0} \frac{1}{(x + h - 1) (x - 1)}$

Step 5.2

Move the term $\frac{1}{x - 1}$ outside of the limit because it is constant with respect to $h$ .

$- \frac{1}{x - 1} lim_{h \to 0} \frac{1}{x + h - 1}$

Step 5.3

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$- \frac{1}{x - 1} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} x + h - 1}$

Step 5.4

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x - 1} \cdot \frac{1}{lim_{h \to 0} x + h - 1}$

Step 5.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$- \frac{1}{x - 1} \cdot \frac{1}{lim_{h \to 0} x + lim_{h \to 0} h - lim_{h \to 0} 1}$

Step 5.6

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x - 1} \cdot \frac{1}{x + lim_{h \to 0} h - lim_{h \to 0} 1}$

Step 5.7

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x - 1} \cdot \frac{1}{x + lim_{h \to 0} h - 1 \cdot 1}$

Step 6

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- \frac{1}{x - 1} \cdot \frac{1}{x + 0 - 1 \cdot 1}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $- 1$ by $1$ .

$- \frac{1}{x - 1} \cdot \frac{1}{x + 0 - 1}$

Step 7.1.2

Add $x$ and $0$ .

$- \frac{1}{x - 1} \cdot \frac{1}{x - 1}$

Step 7.2

Multiply $- \frac{1}{x - 1} \cdot \frac{1}{x - 1}$ .

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Step 7.2.1

Multiply $\frac{1}{x - 1}$ by $\frac{1}{x - 1}$ .

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

Step 7.2.2

Raise $x - 1$ to the power of $1$ .

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

Step 7.2.3

Raise $x - 1$ to the power of $1$ .

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

Step 7.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2.5

Add $1$ and $1$ .

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

Step 8