Calculus Examples

f (x) = \sqrt{x + 2}

$f (x) = \sqrt{x + 2}$

Step 1

Find the first derivative of the function.

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Step 1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x + 2}$ as

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

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

Step 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

$g (x) = x + 2$ .

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Step 1.2.1

To apply the Chain Rule, set

$u$ as

$x + 2$ .

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

Step 1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

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

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

Step 1.2.3

Replace all occurrences of

$u$ with

$x + 2$ .

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

Step 1.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply

$- 1$ by

$2$ .

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

Step 1.6.2

Subtract

$2$ from

$1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

Combine

$\frac{1}{2}$ and

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

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

Step 1.7.3

Move

${(x + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8

By the Sum Rule, the derivative of

$x + 2$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 1.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.10

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

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

Step 1.11

Simplify the expression.

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Step 1.11.1

Add

$1$ and

$0$ .

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

Step 1.11.2

Multiply

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

$1$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since

$\frac{1}{2}$ is constant with respect to

$x$ , the derivative of

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

$x$ is

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

$f'' (x) = \frac{1}{2} \cdot \frac{d}{d x} (\frac{1}{{(x + 2)}^{\frac{1}{2}}})$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite

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

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

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

Step 2.1.2.2

Multiply the exponents in

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

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Step 2.1.2.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2.2.2

Combine

$\frac{1}{2}$ and

$- 1$ .

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

Step 2.1.2.2.3

Move the negative in front of the fraction.

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

Step 2.2

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

$g (x) = x + 2$ .

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Step 2.2.1

To apply the Chain Rule, set

$u$ as

$x + 2$ .

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

Step 2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

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

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

Step 2.2.3

Replace all occurrences of

$u$ with

$x + 2$ .

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

Step 2.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 2.4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply

$- 1$ by

$2$ .

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

Step 2.6.2

Subtract

$2$ from

$- 1$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2} \cdot {(x + 2)}^{\frac{- 3}{2}} \frac{d}{d x} (x + 2))$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2} \cdot {(x + 2)}^{- \frac{3}{2}} \frac{d}{d x} (x + 2))$

Step 2.7.2

Combine

${(x + 2)}^{- \frac{3}{2}}$ and

$\frac{1}{2}$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{{(x + 2)}^{- \frac{3}{2}}}{2} \cdot \frac{d}{d x} (x + 2))$

Step 2.7.3

Move

${(x + 2)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2 {(x + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (x + 2))$

Step 2.7.4

Multiply

$\frac{1}{2}$ by

$\frac{1}{2 {(x + 2)}^{\frac{3}{2}}}$ .

$f'' (x) = - \frac{1}{2 (2 {(x + 2)}^{\frac{3}{2}})} \cdot \frac{d}{d x} (x + 2)$

Step 2.7.5

Multiply

$2$ by

$2$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (x + 2)$

Step 2.8

By the Sum Rule, the derivative of

$x + 2$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (2))$

Step 2.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}} \cdot (1 + \frac{d}{d x} (2))$

Step 2.10

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}} \cdot (1 + 0)$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add

$1$ and

$0$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}} \cdot 1$

Step 2.11.2

Multiply

$- 1$ by

$1$ .

$f'' (x) = - \frac{1}{4 {(x + 2)}^{\frac{3}{2}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to

$0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x + 2}$ as

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

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

Step 4.1.2

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

$g (x) = x + 2$ .

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Step 4.1.2.1

To apply the Chain Rule, set

$u$ as

$x + 2$ .

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

Step 4.1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

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

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

Step 4.1.2.3

Replace all occurrences of

$u$ with

$x + 2$ .

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

Step 4.1.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 4.1.4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply

$- 1$ by

$2$ .

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

Step 4.1.6.2

Subtract

$2$ from

$1$ .

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

Step 4.1.7

Combine fractions.

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Step 4.1.7.1

Move the negative in front of the fraction.

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

Step 4.1.7.2

Combine

$\frac{1}{2}$ and

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

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

Step 4.1.7.3

Move

${(x + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.1.8

By the Sum Rule, the derivative of

$x + 2$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 4.1.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 4.1.10

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

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

Step 4.1.11

Simplify the expression.

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Step 4.1.11.1

Add

$1$ and

$0$ .

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

Step 4.1.11.2

Multiply

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

$1$ .

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

Step 4.2

The first derivative of

$f (x)$ with respect to

$x$ is

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

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

Step 5

Set the first derivative equal to

$0$ then solve the equation

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

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Step 5.1

Set the first derivative equal to

$0$ .

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

Step 5.2

Set the numerator equal to zero.

$1 = 0$

Step 5.3

Since

$1 \neq 0$ , there are no solutions.

No solution

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule

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

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

Step 6.1.2

Anything raised to

$1$ is the base itself.

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

Step 6.2

Set the denominator in

$\frac{1}{2 \sqrt{x + 2}}$ equal to

$0$ to find where the expression is undefined.

$2 \sqrt{x + 2} = 0$

Step 6.3

Solve for

$x$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x + 2}$ as

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify

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

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Step 6.3.2.2.1.1

Apply the product rule to

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

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

Step 6.3.2.2.1.2

Raise

$2$ to the power of

$2$ .

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

Step 6.3.2.2.1.3

Multiply the exponents in

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

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Step 6.3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.3.2.2.1.3.2

Cancel the common factor of

$2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.4

Simplify.

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

Step 6.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 2 = 0^{2}$

Step 6.3.2.2.1.6

Multiply

$4$ by

$2$ .

$4 x + 8 = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$4 x + 8 = 0$

Step 6.3.3

Solve for

$x$ .

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Step 6.3.3.1

Subtract

$8$ from both sides of the equation.

$4 x = - 8$

Step 6.3.3.2

Divide each term in

$4 x = - 8$ by

$4$ and simplify.

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Step 6.3.3.2.1

Divide each term in

$4 x = - 8$ by

$4$ .

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of

$4$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 6.3.3.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 8}{4}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide

$- 8$ by

$4$ .

$x = - 2$

Step 6.4

Set the radicand in

$\sqrt{x + 2}$ less than

$0$ to find where the expression is undefined.

$x + 2 < 0$

Step 6.5

Subtract

$2$ from both sides of the inequality.

$x < - 2$

Step 6.6

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

$0$ .

$x \leq - 2$

$(- \infty, - 2]$

$x \leq - 2$

$(- \infty, - 2]$

Step 7

Critical points to evaluate.

$x = - 2$

Step 8

Evaluate the second derivative at

$x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1}{4 {((- 2) + 2)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the expression.

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Step 9.1.1

Add

$- 2$ and

$2$ .

$- \frac{1}{4 \cdot 0^{\frac{3}{2}}}$

Step 9.1.2

Rewrite

$0$ as

$0^{2}$ .

$- \frac{1}{4 \cdot {(0^{2})}^{\frac{3}{2}}}$

Step 9.1.3

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 9.2

Cancel the common factor of

$2$ .

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Step 9.2.1

Cancel the common factor.

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 9.2.2

Rewrite the expression.

$- \frac{1}{4 \cdot 0^{3}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Raising

$0$ to any positive power yields

$0$ .

$- \frac{1}{4 \cdot 0}$

Step 9.3.2

Multiply

$4$ by

$0$ .

$- \frac{1}{0}$

Step 9.3.3

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$- \frac{1}{0}$

Step 9.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11