Calculus Examples

$f (x) = \frac{1}{x^{2} - 4}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Rewrite $\frac{1}{x^{2} - 4}$ as ${(x^{2} - 4)}^{- 1}$ .

$f' (x) = \frac{d}{d x} ({(x^{2} - 4)}^{- 1})$

Step 1.1.2

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 $g (x) = x^{2} - 4$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$f' (x) = \frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2} - 4)$

Step 1.1.2.2

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

$f' (x) = - u^{- 2} \frac{d}{d x} (x^{2} - 4)$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

$f' (x) = - {(x^{2} - 4)}^{- 2} \frac{d}{d x} (x^{2} - 4)$

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$f' (x) = - {(x^{2} - 4)}^{- 2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4))$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = - {(x^{2} - 4)}^{- 2} (2 x + \frac{d}{d x} (- 4))$

Step 1.1.3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = - {(x^{2} - 4)}^{- 2} (2 x + 0)$

Step 1.1.3.4

Simplify the expression.

Tap for more steps...

Step 1.1.3.4.1

Add $2 x$ and $0$ .

$f' (x) = - {(x^{2} - 4)}^{- 2} (2 x)$

Step 1.1.3.4.2

Multiply $2$ by $- 1$ .

$f' (x) = - 2 {(x^{2} - 4)}^{- 2} x$

Step 1.1.4

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

$f' (x) = - 2 \frac{1}{{(x^{2} - 4)}^{2}} \cdot x$

Step 1.1.5

Combine terms.

Tap for more steps...

Step 1.1.5.1

Combine $- 2$ and $\frac{1}{{(x^{2} - 4)}^{2}}$ .

$f' (x) = \frac{- 2}{{(x^{2} - 4)}^{2}} \cdot x$

Step 1.1.5.2

Move the negative in front of the fraction.

$f' (x) = - \frac{2}{{(x^{2} - 4)}^{2}} \cdot x$

Step 1.1.5.3

Combine $x$ and $\frac{2}{{(x^{2} - 4)}^{2}}$ .

$f' (x) = - \frac{x \cdot 2}{{(x^{2} - 4)}^{2}}$

Step 1.1.5.4

Move $2$ to the left of $x$ .

$f' (x) = - \frac{2 x}{{(x^{2} - 4)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 x}{{(x^{2} - 4)}^{2}}$ .

$- \frac{2 x}{{(x^{2} - 4)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{2 x}{{(x^{2} - 4)}^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{2 x}{{(x^{2} - 4)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 x = 0$

Step 2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 4

Find where the derivative is undefined.

Tap for more steps...

Step 4.1

Set the denominator in $\frac{2 x}{{(x^{2} - 4)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 4)}^{2} = 0$

Step 4.2

Solve for $x$ .

Tap for more steps...

Step 4.2.1

Factor the left side of the equation.

Tap for more steps...

Step 4.2.1.1

Rewrite $4$ as $2^{2}$ .

${(x^{2} - 2^{2})}^{2} = 0$

Step 4.2.1.2

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

${((x + 2) (x - 2))}^{2} = 0$

Step 4.2.1.3

Apply the product rule to $(x + 2) (x - 2)$ .

${(x + 2)}^{2} {(x - 2)}^{2} = 0$

Step 4.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 2)}^{2} = 0$

${(x - 2)}^{2} = 0$

Step 4.2.3

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.2.3.1

Set ${(x + 2)}^{2}$ equal to $0$ .

${(x + 2)}^{2} = 0$

Step 4.2.3.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 4.2.3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.2.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.2.4

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.2.4.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 4.2.4.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 4.2.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.2.5

The final solution is all the values that make ${(x + 2)}^{2} {(x - 2)}^{2} = 0$ true.

$x = - 2, 2$

Step 4.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = - \frac{2 (- 3)}{{({(- 3)}^{2} - 4)}^{2}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $2$ by $- 3$ .

$f' (- 3) = - \frac{- 6}{{({(- 3)}^{2} - 4)}^{2}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Raise $- 3$ to the power of $2$ .

$f' (- 3) = - \frac{- 6}{{(9 - 4)}^{2}}$

Step 6.2.2.2

Subtract $4$ from $9$ .

$f' (- 3) = - \frac{- 6}{5^{2}}$

Step 6.2.2.3

Raise $5$ to the power of $2$ .

$f' (- 3) = - \frac{- 6}{25}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 3) = \frac{6}{25}$

Step 6.2.4

The final answer is $\frac{6}{25}$ .

$\frac{6}{25}$

Step 6.3

At $x = - 3$ the derivative is $\frac{6}{25}$ . Since this is positive, the function is increasing on $(- \infty, - 2)$ .

Increasing on $(- \infty, - 2)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 2, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{2 (- 1)}{{({(- 1)}^{2} - 4)}^{2}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $2$ by $- 1$ .

$f' (- 1) = - \frac{- 2}{{({(- 1)}^{2} - 4)}^{2}}$

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

Raise $- 1$ to the power of $2$ .

$f' (- 1) = - \frac{- 2}{{(1 - 4)}^{2}}$

Step 7.2.2.2

Subtract $4$ from $1$ .

$f' (- 1) = - \frac{- 2}{{(- 3)}^{2}}$

Step 7.2.2.3

Raise $- 3$ to the power of $2$ .

$f' (- 1) = - \frac{- 2}{9}$

Step 7.2.3

Move the negative in front of the fraction.

$f' (- 1) = \frac{2}{9}$

Step 7.2.4

The final answer is $\frac{2}{9}$ .

$\frac{2}{9}$

Step 7.3

At $x = - 1$ the derivative is $\frac{2}{9}$ . Since this is positive, the function is increasing on $(- 2, 0)$ .

Increasing on $(- 2, 0)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(0, 2)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{2 (1)}{{({(1)}^{2} - 4)}^{2}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Multiply $2$ by $1$ .

$f' (1) = - \frac{2}{{(1^{2} - 4)}^{2}}$

Step 8.2.2

Simplify the denominator.

Tap for more steps...

Step 8.2.2.1

One to any power is one.

$f' (1) = - \frac{2}{{(1 - 4)}^{2}}$

Step 8.2.2.2

Subtract $4$ from $1$ .

$f' (1) = - \frac{2}{{(- 3)}^{2}}$

Step 8.2.2.3

Raise $- 3$ to the power of $2$ .

$f' (1) = - \frac{2}{9}$

Step 8.2.3

The final answer is $- \frac{2}{9}$ .

$- \frac{2}{9}$

Step 8.3

At $x = 1$ the derivative is $- \frac{2}{9}$ . Since this is negative, the function is decreasing on $(0, 2)$ .

Decreasing on $(0, 2)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 9.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = - \frac{2 (3)}{{({(3)}^{2} - 4)}^{2}}$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Multiply $2$ by $3$ .

$f' (3) = - \frac{6}{{(3^{2} - 4)}^{2}}$

Step 9.2.2

Simplify the denominator.

Tap for more steps...

Step 9.2.2.1

Raise $3$ to the power of $2$ .

$f' (3) = - \frac{6}{{(9 - 4)}^{2}}$

Step 9.2.2.2

Subtract $4$ from $9$ .

$f' (3) = - \frac{6}{5^{2}}$

Step 9.2.2.3

Raise $5$ to the power of $2$ .

$f' (3) = - \frac{6}{25}$

Step 9.2.3

The final answer is $- \frac{6}{25}$ .

$- \frac{6}{25}$

Step 9.3

At $x = 3$ the derivative is $- \frac{6}{25}$ . Since this is negative, the function is decreasing on $(2, \infty)$ .

Decreasing on $(2, \infty)$ since $f' (x) < 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 2), (- 2, 0)$

Decreasing on: $(0, 2), (2, \infty)$

Step 11