Calculus Examples

$e^{x} (- x^{\frac{2}{3}} - \frac{4}{3} x^{- \frac{3}{2}})$

Step 1

Combine fractions.

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Step 1.1

Combine $x^{- \frac{3}{2}}$ and $\frac{4}{3}$ .

$\frac{d}{d x} [e^{x} (- x^{\frac{2}{3}} - \frac{x^{- \frac{3}{2}} \cdot 4}{3})]$

Step 1.2

Simplify the expression.

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Step 1.2.1

Move $4$ to the left of $x^{- \frac{3}{2}}$ .

$\frac{d}{d x} [e^{x} (- x^{\frac{2}{3}} - \frac{4 \cdot x^{- \frac{3}{2}}}{3})]$

Step 1.2.2

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [e^{x} (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = - x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}$ .

$e^{x} \frac{d}{d x} [- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}] + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 3

Differentiate.

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Step 3.1

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

$e^{x} (\frac{d}{d x} [- x^{\frac{2}{3}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{2}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$e^{x} (- \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$e^{x} (- (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 4

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

$e^{x} (- (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 5

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

$e^{x} (- (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 6

Combine the numerators over the common denominator.

$e^{x} (- (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$e^{x} (- (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 7.2

Subtract $3$ from $2$ .

$e^{x} (- (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$e^{x} (- (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 8.2

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$e^{x} (- \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 8.3

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

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- \frac{4}{3 x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 9

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4}{3 x^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 10

Apply basic rules of exponents.

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Step 10.1

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 10.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 10.2.1

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

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [x^{\frac{3}{2} \cdot - 1}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 10.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 10.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [x^{\frac{3 \cdot - 1}{2}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 10.2.2.2

Multiply $3$ by $- 1$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [x^{\frac{- 3}{2}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 10.2.3

Move the negative in front of the fraction.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} \frac{d}{d x} [x^{- \frac{3}{2}}]) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{2}$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{- \frac{3}{2} - 1})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 12

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

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{- \frac{3}{2} - 1 \cdot \frac{2}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 13

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

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{- \frac{3}{2} + \frac{- 1 \cdot 2}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 14

Combine the numerators over the common denominator.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{\frac{- 3 - 1 \cdot 2}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 15

Simplify the numerator.

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Step 15.1

Multiply $- 1$ by $2$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{\frac{- 3 - 2}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 15.2

Subtract $2$ from $- 3$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{\frac{- 5}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 16

Move the negative in front of the fraction.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{3}{2} x^{- \frac{5}{2}})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 17

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

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} - \frac{4}{3} (- \frac{x^{- \frac{5}{2}} \cdot 3}{2})) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 18

Multiply.

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Step 18.1

Multiply $- 1$ by $- 1$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + 1 (\frac{4}{3}) \frac{x^{- \frac{5}{2}} \cdot 3}{2}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 18.2

Multiply $\frac{4}{3}$ by $1$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{4}{3} \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 19

Multiply $\frac{4}{3}$ by $\frac{x^{- \frac{5}{2}} \cdot 3}{2}$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{4 (x^{- \frac{5}{2}} \cdot 3)}{3 \cdot 2}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 20

Multiply.

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Step 20.1

Multiply $3$ by $4$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{12 x^{- \frac{5}{2}}}{3 \cdot 2}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 20.2

Multiply $3$ by $2$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{12 x^{- \frac{5}{2}}}{6}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 20.3

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{12}{6 x^{\frac{5}{2}}}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 21

Factor $6$ out of $12$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{6 \cdot 2}{6 x^{\frac{5}{2}}}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 22

Cancel the common factors.

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Step 22.1

Factor $6$ out of $6 x^{\frac{5}{2}}$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{6 \cdot 2}{6 (x^{\frac{5}{2}})}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 22.2

Cancel the common factor.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{6 \cdot 2}{6 x^{\frac{5}{2}}}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 22.3

Rewrite the expression.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{x^{\frac{5}{2}}}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) \frac{d}{d x} [e^{x}]$

Step 23

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{x^{\frac{5}{2}}}) + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) e^{x}$

Step 24

Simplify.

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Step 24.1

Apply the distributive property.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}}) + e^{x} \frac{2}{x^{\frac{5}{2}}} + (- x^{\frac{2}{3}} - \frac{4}{3 x^{\frac{3}{2}}}) e^{x}$

Step 24.2

Apply the distributive property.

$e^{x} (- \frac{2}{3 x^{\frac{1}{3}}}) + e^{x} \frac{2}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4}{3 x^{\frac{3}{2}}} e^{x}$

Step 24.3

Combine terms.

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Step 24.3.1

Combine $e^{x}$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$- \frac{e^{x} \cdot 2}{3 x^{\frac{1}{3}}} + e^{x} \frac{2}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4}{3 x^{\frac{3}{2}}} e^{x}$

Step 24.3.2

Move $2$ to the left of $e^{x}$ .

$- \frac{2 \cdot e^{x}}{3 x^{\frac{1}{3}}} + e^{x} \frac{2}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4}{3 x^{\frac{3}{2}}} e^{x}$

Step 24.3.3

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

$- \frac{2 e^{x}}{3 x^{\frac{1}{3}}} + \frac{e^{x} \cdot 2}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4}{3 x^{\frac{3}{2}}} e^{x}$

Step 24.3.4

Move $2$ to the left of $e^{x}$ .

$- \frac{2 e^{x}}{3 x^{\frac{1}{3}}} + \frac{2 \cdot e^{x}}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4}{3 x^{\frac{3}{2}}} e^{x}$

Step 24.3.5

Combine $e^{x}$ and $\frac{4}{3 x^{\frac{3}{2}}}$ .

$- \frac{2 e^{x}}{3 x^{\frac{1}{3}}} + \frac{2 e^{x}}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{e^{x} \cdot 4}{3 x^{\frac{3}{2}}}$

Step 24.3.6

Move $4$ to the left of $e^{x}$ .

$- \frac{2 e^{x}}{3 x^{\frac{1}{3}}} + \frac{2 e^{x}}{x^{\frac{5}{2}}} - x^{\frac{2}{3}} e^{x} - \frac{4 e^{x}}{3 x^{\frac{3}{2}}}$

Step 24.4

Reorder terms.

$- e^{x} x^{\frac{2}{3}} + \frac{2 e^{x}}{x^{\frac{5}{2}}} - \frac{2 e^{x}}{3 x^{\frac{1}{3}}} - \frac{4 e^{x}}{3 x^{\frac{3}{2}}}$