Calculus Examples

$\frac{2}{3} x^{\frac{3}{2}} - \sqrt[3]{4} x + \frac{2}{x^{2}}$

Step 1

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

$\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [- \sqrt[3]{4} x] + \frac{d}{d x} [\frac{2}{x^{2}}]$

Step 2

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

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Step 2.1

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

$\frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- \sqrt[3]{4} x] + \frac{d}{d x} [\frac{2}{x^{2}}]$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $3$ .

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

Step 2.7

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

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

Step 2.8

Multiply $\frac{2}{3}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

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

Step 2.9

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d x} [- \sqrt[3]{4} x] + \frac{d}{d x} [\frac{2}{x^{2}}]$

Step 2.10

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [- \sqrt[3]{4} x] + \frac{d}{d x} [\frac{2}{x^{2}}]$

Step 2.11

Cancel the common factor.

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [- \sqrt[3]{4} x] + \frac{d}{d x} [\frac{2}{x^{2}}]$

Step 2.12

Divide $x^{\frac{1}{2}}$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $2$ by $- 2$ .

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

Step 4.6

Multiply $2$ by $- 1$ .

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

Step 4.7

Raise $x$ to the power of $1$ .

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

Step 4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.9

Subtract $4$ from $1$ .

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

Step 4.10

Multiply $- 2$ by $2$ .

$x^{\frac{1}{2}} - \sqrt[3]{4} - 4 x^{- 3}$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{1}{2}} - \sqrt[3]{4} - 4 \frac{1}{x^{3}}$

Step 6

Simplify.

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Step 6.1

Combine terms.

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Step 6.1.1

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

$x^{\frac{1}{2}} - \sqrt[3]{4} + \frac{- 4}{x^{3}}$

Step 6.1.2

Move the negative in front of the fraction.

$x^{\frac{1}{2}} - \sqrt[3]{4} - \frac{4}{x^{3}}$

Step 6.2

Reorder terms.

$- \frac{4}{x^{3}} + x^{\frac{1}{2}} - \sqrt[3]{4}$