Calculus Examples

$(3 x + 2 \sqrt{x} + \frac{32}{x^{2}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [3 x + 2 x^{\frac{1}{2}} + \frac{32}{x^{2}}]$

Step 2

By the Sum Rule, the derivative of $3 x + 2 x^{\frac{1}{2}} + \frac{32}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 3

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 4

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

$3 \cdot 1 + \frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 5

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 6

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

$3 + 2 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 7

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

$3 + 2 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 8

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

$3 + 2 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 9

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

$3 + 2 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 10

Combine the numerators over the common denominator.

$3 + 2 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $- 1$ by $2$ .

$3 + 2 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 11.2

Subtract $2$ from $1$ .

$3 + 2 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12

Simplify terms.

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Step 12.1

Move the negative in front of the fraction.

$3 + 2 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12.2

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

$3 + 2 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12.3

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

$3 + \frac{2 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12.4

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

$3 + \frac{2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12.5

Cancel the common factor.

$3 + \frac{2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 12.6

Rewrite the expression.

$3 + \frac{1}{x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 13

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{2}}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$3 + \frac{1}{x^{\frac{1}{2}}} + 32 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 14

Apply basic rules of exponents.

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Step 14.1

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

$3 + \frac{1}{x^{\frac{1}{2}}} + 32 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 14.2

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

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Step 14.2.1

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

$3 + \frac{1}{x^{\frac{1}{2}}} + 32 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 14.2.2

Multiply $2$ by $- 1$ .

$3 + \frac{1}{x^{\frac{1}{2}}} + 32 \frac{d}{d x} [x^{- 2}]$

Step 15

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

$3 + \frac{1}{x^{\frac{1}{2}}} + 32 (- 2 x^{- 3})$

Step 16

Multiply $- 2$ by $32$ .

$3 + \frac{1}{x^{\frac{1}{2}}} - 64 x^{- 3}$

Step 17

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

$3 + \frac{1}{x^{\frac{1}{2}}} - 64 \frac{1}{x^{3}}$

Step 18

Simplify.

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Step 18.1

Combine terms.

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Step 18.1.1

Combine $- 64$ and $\frac{1}{x^{3}}$ .

$3 + \frac{1}{x^{\frac{1}{2}}} + \frac{- 64}{x^{3}}$

Step 18.1.2

Move the negative in front of the fraction.

$3 + \frac{1}{x^{\frac{1}{2}}} - \frac{64}{x^{3}}$

Step 18.2

Reorder terms.

$- \frac{64}{x^{3}} + 3 + \frac{1}{x^{\frac{1}{2}}}$