Calculus Examples

$f (x) = 4 x^{2} + \frac{12}{\sqrt{x^{3}}} + 3$

Step 1

By the Sum Rule, the derivative of $4 x^{2} + \frac{12}{\sqrt{x^{3}}} + 3$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}] + \frac{d}{d x} [3]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1

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

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}] + \frac{d}{d x} [3]$

Step 2.2

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

$4 (2 x) + \frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}] + \frac{d}{d x} [3]$

Step 2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}] + \frac{d}{d x} [3]$

Step 3

Evaluate $\frac{d}{d x} [\frac{12}{\sqrt{x^{3}}}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$8 x + \frac{d}{d x} [\frac{12}{x^{\frac{3}{2}}}] + \frac{d}{d x} [3]$

Step 3.2

Since $12$ is constant with respect to $x$ , the derivative of $\frac{12}{x^{\frac{3}{2}}}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$8 x + 12 \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}] + \frac{d}{d x} [3]$

Step 3.3

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

$8 x + 12 \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}] + \frac{d}{d x} [3]$

Step 3.4

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 $g (x) = x^{\frac{3}{2}}$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{\frac{3}{2}}$ .

$8 x + 12 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [3]$

Step 3.4.2

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

$8 x + 12 (- u^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [3]$

Step 3.4.3

Replace all occurrences of $u$ with $x^{\frac{3}{2}}$ .

$8 x + 12 (- {(x^{\frac{3}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [3]$

Step 3.5

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

$8 x + 12 (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.6

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

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Step 3.6.1

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

$8 x + 12 (- x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.6.2

Cancel the common factor of $2$ .

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Step 3.6.2.1

Factor $2$ out of $- 2$ .

$8 x + 12 (- x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.6.2.2

Cancel the common factor.

$8 x + 12 (- x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.6.2.3

Rewrite the expression.

$8 x + 12 (- x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.6.3

Multiply $3$ by $- 1$ .

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [3]$

Step 3.7

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

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} [3]$

Step 3.8

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

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} [3]$

Step 3.9

Combine the numerators over the common denominator.

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})) + \frac{d}{d x} [3]$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}})) + \frac{d}{d x} [3]$

Step 3.10.2

Subtract $2$ from $3$ .

$8 x + 12 (- x^{- 3} (\frac{3}{2} x^{\frac{1}{2}})) + \frac{d}{d x} [3]$

Step 3.11

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

$8 x + 12 (- x^{- 3} \frac{3 x^{\frac{1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.12

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

$8 x + 12 (- \frac{3 x^{\frac{1}{2}} x^{- 3}}{2}) + \frac{d}{d x} [3]$

Step 3.13

Multiply $x^{\frac{1}{2}}$ by $x^{- 3}$ by adding the exponents.

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Step 3.13.1

Move $x^{- 3}$ .

$8 x + 12 (- \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2}) + \frac{d}{d x} [3]$

Step 3.13.2

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

$8 x + 12 (- \frac{3 x^{- 3 + \frac{1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 x + 12 (- \frac{3 x^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.4

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

$8 x + 12 (- \frac{3 x^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.5

Combine the numerators over the common denominator.

$8 x + 12 (- \frac{3 x^{\frac{- 3 \cdot 2 + 1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.6

Simplify the numerator.

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Step 3.13.6.1

Multiply $- 3$ by $2$ .

$8 x + 12 (- \frac{3 x^{\frac{- 6 + 1}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.6.2

Add $- 6$ and $1$ .

$8 x + 12 (- \frac{3 x^{\frac{- 5}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.13.7

Move the negative in front of the fraction.

$8 x + 12 (- \frac{3 x^{- \frac{5}{2}}}{2}) + \frac{d}{d x} [3]$

Step 3.14

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

$8 x + 12 (- \frac{3}{2 x^{\frac{5}{2}}}) + \frac{d}{d x} [3]$

Step 3.15

Multiply $- 1$ by $12$ .

$8 x - 12 \frac{3}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.16

Combine $- 12$ and $\frac{3}{2 x^{\frac{5}{2}}}$ .

$8 x + \frac{- 12 \cdot 3}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.17

Multiply $- 12$ by $3$ .

$8 x + \frac{- 36}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.18

Factor $2$ out of $- 36$ .

$8 x + \frac{2 \cdot - 18}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.19

Cancel the common factors.

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Step 3.19.1

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

$8 x + \frac{2 \cdot - 18}{2 (x^{\frac{5}{2}})} + \frac{d}{d x} [3]$

Step 3.19.2

Cancel the common factor.

$8 x + \frac{2 \cdot - 18}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.19.3

Rewrite the expression.

$8 x + \frac{- 18}{x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 3.20

Move the negative in front of the fraction.

$8 x - \frac{18}{x^{\frac{5}{2}}} + \frac{d}{d x} [3]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$8 x - \frac{18}{x^{\frac{5}{2}}} + 0$

Step 4.2

Add $8 x - \frac{18}{x^{\frac{5}{2}}}$ and $0$ .

$8 x - \frac{18}{x^{\frac{5}{2}}}$