Calculus Examples

$f (x) = {(x + 3)}^{\frac{2}{5}}$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{5}}$ and $g (x) = x + 3$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$\frac{d}{d u} [u^{\frac{2}{5}}] \frac{d}{d x} [x + 3]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{5}$ .

$\frac{2}{5} u^{\frac{2}{5} - 1} \frac{d}{d x} [x + 3]$

Step 1.1.1.3

Replace all occurrences of $u$ with $x + 3$ .

$\frac{2}{5} {(x + 3)}^{\frac{2}{5} - 1} \frac{d}{d x} [x + 3]$

Step 1.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2}{5} {(x + 3)}^{\frac{2}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.3

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{2}{5} {(x + 3)}^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{2}{5} {(x + 3)}^{\frac{2 - 1 \cdot 5}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $5$ .

$\frac{2}{5} {(x + 3)}^{\frac{2 - 5}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.5.2

Subtract $5$ from $2$ .

$\frac{2}{5} {(x + 3)}^{\frac{- 3}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

$\frac{2}{5} {(x + 3)}^{- \frac{3}{5}} \frac{d}{d x} [x + 3]$

Step 1.1.6.2

Combine $\frac{2}{5}$ and ${(x + 3)}^{- \frac{3}{5}}$ .

$\frac{2 {(x + 3)}^{- \frac{3}{5}}}{5} \frac{d}{d x} [x + 3]$

Step 1.1.6.3

Move ${(x + 3)}^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} \frac{d}{d x} [x + 3]$

Step 1.1.7

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} (\frac{d}{d x} [x] + \frac{d}{d x} [3])$

Step 1.1.8

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

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} (1 + \frac{d}{d x} [3])$

Step 1.1.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} (1 + 0)$

Step 1.1.10

Simplify the expression.

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Step 1.1.10.1

Add $1$ and $0$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} \cdot 1$

Step 1.1.10.2

Multiply $\frac{2}{5 {(x + 3)}^{\frac{3}{5}}}$ by $1$ .

$f' (x) = \frac{2}{5 {(x + 3)}^{\frac{3}{5}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2}{5 {(x + 3)}^{\frac{3}{5}}}$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2}{5 {(x + 3)}^{\frac{3}{5}}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

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

$\frac{2}{5 \sqrt[5]{{(x + 3)}^{3}}}$

Step 4.2

Set the denominator in $\frac{2}{5 \sqrt[5]{{(x + 3)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{{(x + 3)}^{3}} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(5 \sqrt[5]{{(x + 3)}^{3}})}^{5} = 0^{5}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(x + 3)}^{3}}$ as ${(x + 3)}^{\frac{3}{5}}$ .

${(5 {(x + 3)}^{\frac{3}{5}})}^{5} = 0^{5}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(5 {(x + 3)}^{\frac{3}{5}})}^{5}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $5 {(x + 3)}^{\frac{3}{5}}$ .

$5^{5} {({(x + 3)}^{\frac{3}{5}})}^{5} = 0^{5}$

Step 4.3.2.2.1.2

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

$3125 {({(x + 3)}^{\frac{3}{5}})}^{5} = 0^{5}$

Step 4.3.2.2.1.3

Multiply the exponents in ${({(x + 3)}^{\frac{3}{5}})}^{5}$ .

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Step 4.3.2.2.1.3.1

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

$3125 {(x + 3)}^{\frac{3}{5} \cdot 5} = 0^{5}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$3125 {(x + 3)}^{\frac{3}{5} \cdot 5} = 0^{5}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$3125 {(x + 3)}^{3} = 0^{5}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$3125 {(x + 3)}^{3} = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Divide each term in $3125 {(x + 3)}^{3} = 0$ by $3125$ and simplify.

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Step 4.3.3.1.1

Divide each term in $3125 {(x + 3)}^{3} = 0$ by $3125$ .

$\frac{3125 {(x + 3)}^{3}}{3125} = \frac{0}{3125}$

Step 4.3.3.1.2

Simplify the left side.

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Step 4.3.3.1.2.1

Cancel the common factor of $3125$ .

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Step 4.3.3.1.2.1.1

Cancel the common factor.

$\frac{3125 {(x + 3)}^{3}}{3125} = \frac{0}{3125}$

Step 4.3.3.1.2.1.2

Divide ${(x + 3)}^{3}$ by $1$ .

${(x + 3)}^{3} = \frac{0}{3125}$

Step 4.3.3.1.3

Simplify the right side.

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Step 4.3.3.1.3.1

Divide $0$ by $3125$ .

${(x + 3)}^{3} = 0$

Step 4.3.3.2

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.3.3.3

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

After finding the point that makes the derivative $f' (x) = \frac{2}{5 {(x + 3)}^{\frac{3}{5}}}$ equal to $0$ or undefined, the interval to check where $f (x) = {(x + 3)}^{\frac{2}{5}}$ is increasing and where it is decreasing is $(- \infty, - 3) \cup (- 3, \infty)$ .

$(- \infty, - 3) \cup (- 3, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = \frac{2}{5 {((- 4) + 3)}^{\frac{3}{5}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Add $- 4$ and $3$ .

$f' (- 4) = \frac{2}{5 {(- 1)}^{\frac{3}{5}}}$

Step 6.2.1.2

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$f' (- 4) = \frac{2}{5 {({(- 1)}^{5})}^{\frac{3}{5}}}$

Step 6.2.1.3

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

$f' (- 4) = \frac{2}{5 {(- 1)}^{5 (\frac{3}{5})}}$

Step 6.2.1.4

Cancel the common factor of $5$ .

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Step 6.2.1.4.1

Cancel the common factor.

$f' (- 4) = \frac{2}{5 {(- 1)}^{5 (\frac{3}{5})}}$

Step 6.2.1.4.2

Rewrite the expression.

$f' (- 4) = \frac{2}{5 {(- 1)}^{3}}$

Step 6.2.1.5

Raise $- 1$ to the power of $3$ .

$f' (- 4) = \frac{2}{5 \cdot - 1}$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Multiply $5$ by $- 1$ .

$f' (- 4) = \frac{2}{- 5}$

Step 6.2.2.2

Move the negative in front of the fraction.

$f' (- 4) = - \frac{2}{5}$

Step 6.2.3

The final answer is $- \frac{2}{5}$ .

$- \frac{2}{5}$

Step 6.3

At $x = - 4$ the derivative is $- \frac{2}{5}$ . Since this is negative, the function is decreasing on $(- \infty, - 3)$ .

Decreasing on $(- \infty, - 3)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Add $- 2$ and $3$ .

$f' (- 2) = \frac{2}{5 \cdot 1^{\frac{3}{5}}}$

Step 7.2.1.2

One to any power is one.

$f' (- 2) = \frac{2}{5 \cdot 1}$

Step 7.2.2

Multiply $5$ by $1$ .

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

Step 7.2.3

The final answer is $\frac{2}{5}$ .

$\frac{2}{5}$

Step 7.3

At $x = - 2$ the derivative is $\frac{2}{5}$ . Since this is positive, the function is increasing on $(- 3, \infty)$ .

Increasing on $(- 3, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- 3, \infty)$

Decreasing on: $(- \infty, - 3)$

Step 9