Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

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

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Step 4.1.5.1

Expand $(x + h) (x + 1)$ using the FOIL Method.

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Step 4.1.5.1.1

Apply the distributive property.

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

Step 4.1.5.1.2

Apply the distributive property.

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

Step 4.1.5.1.3

Apply the distributive property.

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

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x \cdot 1 + h x + h \cdot 1 - x (x + h + 1)}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.2.2

Multiply $x$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h \cdot 1 - x (x + h + 1)}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.2.3

Multiply $h$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h - x (x + h + 1)}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.3

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h - x \cdot x - x h - x \cdot 1}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.4

Simplify.

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Step 4.1.5.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.4.1.1

Move $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h - (x \cdot x) - x h - x \cdot 1}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.4.1.2

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h - x^{2} - x h - x \cdot 1}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.4.2

Multiply $- 1$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x^{2} + x + h x + h - x^{2} - x h - x}{(x + 1) (x + h + 1)}}{h}$

Step 4.1.5.5

Subtract $x^{2}$ from $x^{2}$ .

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

Step 4.1.5.6

Add $x$ and $0$ .

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

Step 4.1.5.7

Subtract $x$ from $x$ .

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

Step 4.1.5.8

Add $h x + h - x h$ and $0$ .

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

Step 4.1.5.9

Subtract $x h$ from $h x$ .

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Step 4.1.5.9.1

Reorder $h$ and $x$ .

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

Step 4.1.5.9.2

Subtract $x h$ from $x h$ .

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

Step 4.1.5.10

Add $h$ and $0$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 5

Evaluate the limit.

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Step 5.1

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 5.2

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 5.3

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 5.4

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 5.5

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 5.6

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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