Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

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 $g (x) = x^{2} - 1$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

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

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

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.6.2

Subtract $2$ from $1$ .

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

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

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

Step 1.1.7.2

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

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

Step 1.1.7.3

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

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

Step 1.1.8

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.9

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

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

Step 1.1.10

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.1.11

Simplify terms.

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Step 1.1.11.1

Add $2 x$ and $0$ .

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

Step 1.1.11.2

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

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

Step 1.1.11.3

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

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

Step 1.1.11.4

Cancel the common factor.

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

Step 1.1.11.5

Rewrite the expression.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

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

Step 3.1.2

Anything raised to $1$ is the base itself.

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

Step 3.2

Set the denominator in $\frac{x}{\sqrt{x^{2} - 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{2} - 1} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

${\sqrt{x^{2} - 1}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

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Step 3.3.2.2.1.1.1

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

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

Step 3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 3.3.2.2.1.2

Simplify.

$x^{2} - 1 = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$x^{2} - 1 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 3.3.3.2

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

$x = \pm \sqrt{1}$

Step 3.3.3.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 3.3.3.4

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

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Step 3.3.3.4.1

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

$x = 1$

Step 3.3.3.4.2

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

$x = - 1$

Step 3.3.3.4.3

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

$x = 1, - 1$

Step 3.4

Set the radicand in $\sqrt{x^{2} - 1}$ less than $0$ to find where the expression is undefined.

$x^{2} - 1 < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Add $1$ to both sides of the inequality.

$x^{2} < 1$

Step 3.5.2

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

$\sqrt{x^{2}} < \sqrt{1}$

Step 3.5.3

Simplify the equation.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Pull terms out from under the radical.

$| x | < \sqrt{1}$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Any root of $1$ is $1$ .

$| x | < 1$

Step 3.5.4

Write $| x | < 1$ as a piecewise.

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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.

$x < 1$

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 $- 1$ .

$- x < 1$

Step 3.5.4.5

Write as a piecewise.

${\begin{cases} x < 1 & x \geq 0 \\ - x < 1 & x < 0 \end{cases}$

Step 3.5.5

Find the intersection of $x < 1$ and $x \geq 0$ .

$0 \leq x < 1$

Step 3.5.6

Solve $- x < 1$ when $x < 0$ .

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Step 3.5.6.1

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

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Step 3.5.6.1.1

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

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

Step 3.5.6.1.2

Simplify the left side.

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Step 3.5.6.1.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.6.1.2.2

Divide $x$ by $1$ .

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

Step 3.5.6.1.3

Simplify the right side.

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Step 3.5.6.1.3.1

Divide $1$ by $- 1$ .

$x > - 1$

Step 3.5.6.2

Find the intersection of $x > - 1$ and $x < 0$ .

$- 1 < x < 0$

Step 3.5.7

Find the union of the solutions.

$- 1 < x < 1$

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 $0$ .

$- 1 \leq x \leq 1$

$[- 1, 1]$

$- 1 \leq x \leq 1$

$[- 1, 1]$

Step 4

Evaluate $\sqrt{x^{2} - 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\sqrt{{(0)}^{2} - 1}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\sqrt{0 - 1}$

Step 4.1.2.2

Subtract $1$ from $0$ .

$\sqrt{- 1}$

Step 4.1.2.3

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

$i$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

$\sqrt{{(- 1)}^{2} - 1}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\sqrt{1 - 1}$

Step 4.2.2.2

Subtract $1$ from $1$ .

$\sqrt{0}$

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 $x = 1$ .

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Step 4.3.1

Substitute $1$ for $x$ .

$\sqrt{{(1)}^{2} - 1}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

One to any power is one.

$\sqrt{1 - 1}$

Step 4.3.2.2

Subtract $1$ from $1$ .

$\sqrt{0}$

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.

$(- 1, 0), (1, 0)$

Step 5