Calculus Examples

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

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.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

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

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 $1$ .

Tap for more steps...

Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

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

Tap for more steps...

Step 4.1.5.1

Expand $(x + h - 1) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.2

Combine the opposite terms in $x \cdot x + x \cdot 1 + h x + h \cdot 1 - 1 x - 1 \cdot 1$ .

Tap for more steps...

Step 4.1.5.2.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 4.1.5.2.2

Subtract $1 x$ from $1 x$ .

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

Step 4.1.5.2.3

Add $x \cdot x + h x + h \cdot 1$ and $0$ .

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

Step 4.1.5.3

Simplify each term.

Tap for more steps...

Step 4.1.5.3.1

Multiply $x$ by $x$ .

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

Step 4.1.5.3.2

Multiply $h$ by $1$ .

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

Step 4.1.5.3.3

Multiply $- 1$ by $1$ .

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

Step 4.1.5.4

Apply the distributive property.

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

Step 4.1.5.5

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.6

Expand $(- x + 1) (x + h + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.7

Simplify each term.

Tap for more steps...

Step 4.1.5.7.1

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

Tap for more steps...

Step 4.1.5.7.1.1

Move $x$ .

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

Step 4.1.5.7.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.7.2

Multiply $- 1$ by $1$ .

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

Step 4.1.5.7.3

Multiply $x$ by $1$ .

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

Step 4.1.5.7.4

Multiply $h$ by $1$ .

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

Step 4.1.5.7.5

Multiply $1$ by $1$ .

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

Step 4.1.5.8

Combine the opposite terms in $- x^{2} - x h - x + x + h + 1$ .

Tap for more steps...

Step 4.1.5.8.1

Add $- x$ and $x$ .

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

Step 4.1.5.8.2

Add $- x^{2} - x h$ and $0$ .

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

Step 4.1.5.9

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

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

Step 4.1.5.10

Add $h x$ and $0$ .

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

Step 4.1.5.11

Subtract $x h$ from $h x$ .

Tap for more steps...

Step 4.1.5.11.1

Reorder $h$ and $x$ .

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

Step 4.1.5.11.2

Subtract $x h$ from $x h$ .

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

Step 4.1.5.12

Add $h$ and $0$ .

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

Step 4.1.5.13

Add $h$ and $h$ .

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

Step 4.1.5.14

Add $- 1$ and $1$ .

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

Step 4.1.5.15

Add $2 h$ and $0$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.3.1

Factor $h$ out of $2 h$ .

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Move the term $\frac{2}{x + 1}$ outside of the limit because it is constant with respect to $h$ .

$\frac{2}{x + 1} lim_{h \to 0} \frac{1}{x + h + 1}$

Step 5.2

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

$\frac{2}{x + 1} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} x + h + 1}$

Step 5.3

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

$\frac{2}{x + 1} \cdot \frac{1}{lim_{h \to 0} x + h + 1}$

Step 5.4

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

$\frac{2}{x + 1} \cdot \frac{1}{lim_{h \to 0} x + lim_{h \to 0} h + lim_{h \to 0} 1}$

Step 5.5

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

$\frac{2}{x + 1} \cdot \frac{1}{x + lim_{h \to 0} h + lim_{h \to 0} 1}$

Step 5.6

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

$\frac{2}{x + 1} \cdot \frac{1}{x + lim_{h \to 0} h + 1}$

Step 6

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

$\frac{2}{x + 1} \cdot \frac{1}{x + 0 + 1}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Add $x$ and $0$ .

$\frac{2}{x + 1} \cdot \frac{1}{x + 1}$

Step 7.2

Multiply $\frac{2}{x + 1} \cdot \frac{1}{x + 1}$ .

Tap for more steps...

Step 7.2.1

Multiply $\frac{2}{x + 1}$ by $\frac{1}{x + 1}$ .

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

Step 7.2.2

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

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

Step 7.2.3

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

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

Step 7.2.4

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

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

Step 7.2.5

Add $1$ and $1$ .

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

Step 8