Calculus Examples

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

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.

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

Step 2.1.2

The final answer is $\frac{1}{{(x + h)}^{2}}$ .

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

Step 2.2

Find the components of the definition.

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

$f (x) = \frac{1}{x^{2}}$

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

$f (x) = \frac{1}{x^{2}}$

Step 3

Plug in the components.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{x + h}$ and $b = \frac{1}{x}$ .

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

Step 4.1.5

Simplify.

Tap for more steps...

Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

Tap for more steps...

Step 4.1.5.3.1

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

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

Step 4.1.5.3.2

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

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

Step 4.1.5.3.3

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

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

Step 4.1.5.4

Combine the numerators over the common denominator.

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

Step 4.1.5.5

Add $x$ and $x$ .

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

Step 4.1.5.6

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

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

Step 4.1.5.7

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

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

Step 4.1.5.8

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

Tap for more steps...

Step 4.1.5.8.1

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

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

Step 4.1.5.8.2

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

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

Step 4.1.5.8.3

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

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

Step 4.1.5.9

Combine the numerators over the common denominator.

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

Step 4.1.5.10

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

Tap for more steps...

Step 4.1.5.10.1

Apply the distributive property.

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

Step 4.1.5.10.2

Subtract $x$ from $x$ .

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

Step 4.1.5.10.3

Subtract $h$ from $0$ .

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

Step 4.1.6

Move the negative in front of the fraction.

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

Step 4.1.7

Combine exponents.

Tap for more steps...

Step 4.1.7.1

Factor out negative.

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

Step 4.1.7.2

Multiply $\frac{2 x + h}{x (x + h)}$ by $\frac{h}{x (x + h)}$ .

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

Step 4.1.7.3

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

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

Step 4.1.7.4

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

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

Step 4.1.7.5

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

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

Step 4.1.7.6

Add $1$ and $1$ .

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

Step 4.1.7.7

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

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

Step 4.1.7.8

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

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

Step 4.1.7.9

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

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

Step 4.1.7.10

Add $1$ and $1$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.3.1

Move the leading negative in $- \frac{(2 x + h) h}{x^{2} {(x + h)}^{2}}$ into the numerator.

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

Step 4.3.2

Factor $h$ out of $- (2 x + h) h$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

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

Step 10

Move the exponent $2$ from ${(x + h)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 11

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

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

Step 12

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

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

Step 13

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

Tap for more steps...

Step 13.1

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

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

Step 13.2

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

$- \frac{1}{x^{2}} \cdot \frac{2 x + 0}{{(x + 0)}^{2}}$

Step 14

Simplify the answer.

Tap for more steps...

Step 14.1

Add $2 x$ and $0$ .

$- \frac{1}{x^{2}} \cdot \frac{2 x}{{(x + 0)}^{2}}$

Step 14.2

Add $x$ and $0$ .

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

Step 14.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 14.3.1

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

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

Step 14.3.2

Factor $x$ out of $x^{2}$ .

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

Step 14.3.3

Factor $x$ out of $2 x$ .

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

Step 14.3.4

Cancel the common factor.

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

Step 14.3.5

Rewrite the expression.

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

Step 14.4

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

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

Step 14.5

Multiply $- 1$ by $2$ .

$\frac{- 2}{x \cdot x^{2}}$

Step 14.6

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 14.6.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 14.6.1.1

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

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

Step 14.6.1.2

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

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

Step 14.6.2

Add $1$ and $2$ .

$\frac{- 2}{x^{3}}$

Step 14.7

Move the negative in front of the fraction.

$- \frac{2}{x^{3}}$

Step 15