Calculus Examples

$2 x^{\frac{5}{3}} - 5 x^{\frac{4}{3}}$

Step 1

Write $2 x^{\frac{5}{3}} - 5 x^{\frac{4}{3}}$ as a function.

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

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $2 x^{\frac{5}{3}} - 5 x^{\frac{4}{3}}$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{5}{3}}] + \frac{d}{d x} [- 5 x^{\frac{4}{3}}]$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [2 x^{\frac{5}{3}}]$ .

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Step 2.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{5}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{5}{3}}]$ .

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

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{3}$ .

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

Step 2.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.1.2.4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.1.2.5

Combine the numerators over the common denominator.

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

Step 2.1.2.6

Simplify the numerator.

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Step 2.1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.6.2

Subtract $3$ from $5$ .

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

Step 2.1.2.7

Combine $\frac{5}{3}$ and $x^{\frac{2}{3}}$ .

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

Step 2.1.2.8

Combine $2$ and $\frac{5 x^{\frac{2}{3}}}{3}$ .

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

Step 2.1.2.9

Multiply $5$ by $2$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} + \frac{d}{d x} (- 5 x^{\frac{4}{3}})$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 5 x^{\frac{4}{3}}]$ .

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Step 2.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{\frac{4}{3}}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{\frac{4}{3}}]$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 \frac{d}{d x} x^{\frac{4}{3}}$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{4}{3} - 1})$

Step 2.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}})$

Step 2.1.3.4

Combine $- 1$ and $\frac{3}{3}$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.1.3.5

Combine the numerators over the common denominator.

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}})$

Step 2.1.3.6

Simplify the numerator.

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Step 2.1.3.6.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{4 - 3}{3}})$

Step 2.1.3.6.2

Subtract $3$ from $4$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 (\frac{4}{3} x^{\frac{1}{3}})$

Step 2.1.3.7

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - 5 \frac{4 x^{\frac{1}{3}}}{3}$

Step 2.1.3.8

Combine $- 5$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} + \frac{- 5 (4 x^{\frac{1}{3}})}{3}$

Step 2.1.3.9

Multiply $4$ by $- 5$ .

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} + \frac{- 20 x^{\frac{1}{3}}}{3}$

Step 2.1.3.10

Move the negative in front of the fraction.

$f' (x) = \frac{10 x^{\frac{2}{3}}}{3} - \frac{20 x^{\frac{1}{3}}}{3}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{10 x^{\frac{2}{3}}}{3} - \frac{20 x^{\frac{1}{3}}}{3}$ .

$\frac{10 x^{\frac{2}{3}}}{3} - \frac{20 x^{\frac{1}{3}}}{3}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{10 x^{\frac{2}{3}}}{3} - \frac{20 x^{\frac{1}{3}}}{3} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{10 x^{\frac{2}{3}}}{3} - \frac{20 x^{\frac{1}{3}}}{3} = 0$

Step 3.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 0, 8$

Step 4

The values which make the derivative equal to $0$ are $0, 8$ .

$0, 8$

Step 5

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

$(- \infty, 0) \cup (0, 8) \cup (8, \infty)$

Step 6

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

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Step 6.1

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

$f' (- 1) = \frac{10 {(- 1)}^{\frac{2}{3}}}{3} - \frac{20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2

Simplify the result.

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Step 6.2.1

Combine the numerators over the common denominator.

$f' (- 1) = \frac{10 {(- 1)}^{\frac{2}{3}} - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f' (- 1) = \frac{10 {({(- 1)}^{3})}^{\frac{2}{3}} - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.2

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

$f' (- 1) = \frac{10 {(- 1)}^{3 (\frac{2}{3})} - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.3

Cancel the common factor of $3$ .

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Step 6.2.2.3.1

Cancel the common factor.

$f' (- 1) = \frac{10 {(- 1)}^{3 (\frac{2}{3})} - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.3.2

Rewrite the expression.

$f' (- 1) = \frac{10 {(- 1)}^{2} - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.4

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

$f' (- 1) = \frac{10 \cdot 1 - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.5

Multiply $10$ by $1$ .

$f' (- 1) = \frac{10 - 20 {(- 1)}^{\frac{1}{3}}}{3}$

Step 6.2.2.6

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f' (- 1) = \frac{10 - 20 {({(- 1)}^{3})}^{\frac{1}{3}}}{3}$

Step 6.2.2.7

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

$f' (- 1) = \frac{10 - 20 {(- 1)}^{3 (\frac{1}{3})}}{3}$

Step 6.2.2.8

Cancel the common factor of $3$ .

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Step 6.2.2.8.1

Cancel the common factor.

$f' (- 1) = \frac{10 - 20 {(- 1)}^{3 (\frac{1}{3})}}{3}$

Step 6.2.2.8.2

Rewrite the expression.

$f' (- 1) = \frac{10 - 20 \cdot - 1}{3}$

Step 6.2.2.9

Evaluate the exponent.

$f' (- 1) = \frac{10 - 20 \cdot - 1}{3}$

Step 6.2.2.10

Multiply $- 20$ by $- 1$ .

$f' (- 1) = \frac{10 + 20}{3}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Add $10$ and $20$ .

$f' (- 1) = \frac{30}{3}$

Step 6.2.3.2

Divide $30$ by $3$ .

$f' (- 1) = 10$

Step 6.2.4

The final answer is $10$ .

$10$

Step 6.3

At $x = - 1$ the derivative is $10$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

Step 7

Substitute a value from the interval $(0, 8)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (4) = \frac{10 {(4)}^{\frac{2}{3}}}{3} - \frac{20 {(4)}^{\frac{1}{3}}}{3}$

Step 7.2

Simplify the result.

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Step 7.2.1

Combine the numerators over the common denominator.

$f' (4) = \frac{10 {(4)}^{\frac{2}{3}} - 20 {(4)}^{\frac{1}{3}}}{3}$

Step 7.2.2

Remove parentheses.

$f' (4) = \frac{10 \cdot 4^{\frac{2}{3}} - 20 \cdot 4^{\frac{1}{3}}}{3}$

Step 7.2.3

The final answer is $\frac{10 \cdot 4^{\frac{2}{3}} - 20 \cdot 4^{\frac{1}{3}}}{3}$ .

$\frac{10 \cdot 4^{\frac{2}{3}} - 20 \cdot 4^{\frac{1}{3}}}{3}$

Step 7.3

At $x = 4$ the derivative is $\frac{10 \cdot 4^{\frac{2}{3}} - 20 \cdot 4^{\frac{1}{3}}}{3}$ . Since this is negative, the function is decreasing on $(0, 8)$ .

Decreasing on $(0, 8)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(8, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = \frac{10 {(9)}^{\frac{2}{3}}}{3} - \frac{20 {(9)}^{\frac{1}{3}}}{3}$

Step 8.2

Simplify the result.

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Step 8.2.1

Combine the numerators over the common denominator.

$f' (9) = \frac{10 {(9)}^{\frac{2}{3}} - 20 {(9)}^{\frac{1}{3}}}{3}$

Step 8.2.2

Remove parentheses.

$f' (9) = \frac{10 \cdot 9^{\frac{2}{3}} - 20 \cdot 9^{\frac{1}{3}}}{3}$

Step 8.2.3

The final answer is $\frac{10 \cdot 9^{\frac{2}{3}} - 20 \cdot 9^{\frac{1}{3}}}{3}$ .

$\frac{10 \cdot 9^{\frac{2}{3}} - 20 \cdot 9^{\frac{1}{3}}}{3}$

Step 8.3

At $x = 9$ the derivative is $\frac{10 \cdot 9^{\frac{2}{3}} - 20 \cdot 9^{\frac{1}{3}}}{3}$ . Since this is negative, the function is decreasing on $(8, \infty)$ .

Decreasing on $(8, \infty)$ since $f' (x) < 0$

Step 9

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

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

Decreasing on: $(0, 8), (8, \infty)$

Step 10