Calculus Examples

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]$

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

Tap for more steps...

Step 1.1.2.1

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

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} - 4]$

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 $n = \frac{1}{2}$ .

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.2.3

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

$\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.6

Simplify the numerator.

Tap for more steps...

Step 1.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} - 4)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.7

Combine fractions.

Tap for more steps...

Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} - 4)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.7.2

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

$\frac{{(x^{2} - 4)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.7.3

Move ${(x^{2} - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.8

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

$\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])$

Step 1.1.9

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

$\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [- 4])$

Step 1.1.10

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

$\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + 0)$

Step 1.1.11

Simplify terms.

Tap for more steps...

Step 1.1.11.1

Add $2 x$ and $0$ .

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

Step 1.1.11.2

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

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

Step 1.1.11.3

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

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

Step 1.1.11.4

Cancel the common factor.

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

Step 1.1.11.5

Rewrite the expression.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{{(x^{2} - 4)}^{\frac{1}{2}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x}{{(x^{2} - 4)}^{\frac{1}{2}}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{x}{{(x^{2} - 4)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$x = 0$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{x}{\sqrt{x^{2} - 4}}$

Step 3.2

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

$\sqrt{x^{2} - 4} = 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 4}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

${({(x^{2} - 4)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

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

Tap for more steps...

Step 3.3.2.2.1.1

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 4)}^{1} = 0^{2}$

Step 3.3.2.2.1.2

Simplify.

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

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{2} - 4 = 0$

Step 3.3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.3.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 3.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 3.3.3.3

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 3.3.3.3.1

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

$x = \pm \sqrt{2^{2}}$

Step 3.3.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 3.3.3.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.3.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 3.3.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 3.3.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 3.4

Set the radicand in $\sqrt{x^{2} - 4}$ less than $0$ to find where the expression is undefined.

$x^{2} - 4 < 0$

Step 3.5

Solve for $x$ .

Tap for more steps...

Step 3.5.1

Add $4$ to both sides of the inequality.

$x^{2} < 4$

Step 3.5.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} < \sqrt{4}$

Step 3.5.3

Simplify the equation.

Tap for more steps...

Step 3.5.3.1

Simplify the left side.

Tap for more steps...

Step 3.5.3.1.1

Pull terms out from under the radical.

$| x | < \sqrt{4}$

Step 3.5.3.2

Simplify the right side.

Tap for more steps...

Step 3.5.3.2.1

Simplify $\sqrt{4}$ .

Tap for more steps...

Step 3.5.3.2.1.1

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

$| x | < \sqrt{2^{2}}$

Step 3.5.3.2.1.2

Pull terms out from under the radical.

$| x | < | 2 |$

Step 3.5.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | < 2$

Step 3.5.4

Write $| x | < 2$ as a piecewise.

Tap for more steps...

Step 3.5.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.5.4.2

In the piece where $x$ is non-negative, remove the absolute value.

$x < 2$

Step 3.5.4.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.5.4.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 2$

Step 3.5.4.5

Write as a piecewise.

${\begin{cases} x < 2 & x \geq 0 \\ - x < 2 & x < 0 \end{cases}$

Step 3.5.5

Find the intersection of $x < 2$ and $x \geq 0$ .

$0 \leq x < 2$

Step 3.5.6

Solve $- x < 2$ when $x < 0$ .

Tap for more steps...

Step 3.5.6.1

Divide each term in $- x < 2$ by $- 1$ and simplify.

Tap for more steps...

Step 3.5.6.1.1

Divide each term in $- x < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.5.6.1.2

Simplify the left side.

Tap for more steps...

Step 3.5.6.1.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.6.1.2.2

Divide $x$ by $1$ .

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

Step 3.5.6.1.3

Simplify the right side.

Tap for more steps...

Step 3.5.6.1.3.1

Divide $2$ by $- 1$ .

$x > - 2$

Step 3.5.6.2

Find the intersection of $x > - 2$ and $x < 0$ .

$- 2 < x < 0$

Step 3.5.7

Find the union of the solutions.

$- 2 < x < 2$

Step 3.6

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 $0$ .

$- 2 \leq x \leq 2$

$[- 2, 2]$

$- 2 \leq x \leq 2$

$[- 2, 2]$

Step 4

Evaluate $\sqrt{x^{2} - 4}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$\sqrt{{(0)}^{2} - 4}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify the expression.

Tap for more steps...

Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 - 4}$

Step 4.1.2.1.2

Subtract $4$ from $0$ .

$\sqrt{- 4}$

Step 4.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$\sqrt{- 1 (4)}$

Step 4.1.2.3

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\sqrt{- 1} \cdot \sqrt{4}$

Step 4.1.2.4

Rewrite $\sqrt{- 1}$ as $i$ .

$i \cdot \sqrt{4}$

Step 4.1.2.5

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

$i \cdot \sqrt{2^{2}}$

Step 4.1.2.6

Pull terms out from under the radical, assuming positive real numbers.

$i \cdot 2$

Step 4.1.2.7

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

$2 i$

Step 4.2

Evaluate at $x = - 2$ .

Tap for more steps...

Step 4.2.1

Substitute $- 2$ for $x$ .

$\sqrt{{(- 2)}^{2} - 4}$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

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

$\sqrt{4 - 4}$

Step 4.2.2.2

Subtract $4$ from $4$ .

$\sqrt{0}$

Step 4.2.2.3

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}}$

Step 4.2.2.4

Pull terms out from under the radical, assuming positive real numbers.

$0$

Step 4.3

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.3.1

Substitute $2$ for $x$ .

$\sqrt{{(2)}^{2} - 4}$

Step 4.3.2

Simplify.

Tap for more steps...

Step 4.3.2.1

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

$\sqrt{4 - 4}$

Step 4.3.2.2

Subtract $4$ from $4$ .

$\sqrt{0}$

Step 4.3.2.3

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}}$

Step 4.3.2.4

Pull terms out from under the radical, assuming positive real numbers.

$0$

Step 4.4

List all of the points.

$(- 2, 0), (2, 0)$

Step 5