Calculus Examples

$h (x) = 12 x - x^{3} - \frac{3}{\sqrt{x^{3}}}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$12 \cdot 1 + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{3}}}]$

Step 2.3

Multiply $12$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $- 1$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

$12 - 3 x^{2} - 3 (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.6

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

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Step 4.6.1

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

$12 - 3 x^{2} - 3 (- x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.6.2

Cancel the common factor of $2$ .

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Step 4.6.2.1

Factor $2$ out of $- 2$ .

$12 - 3 x^{2} - 3 (- x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.6.2.2

Cancel the common factor.

$12 - 3 x^{2} - 3 (- x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.6.2.3

Rewrite the expression.

$12 - 3 x^{2} - 3 (- x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.6.3

Multiply $3$ by $- 1$ .

$12 - 3 x^{2} - 3 (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1}))$

Step 4.7

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

$12 - 3 x^{2} - 3 (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}))$

Step 4.8

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

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

Step 4.9

Combine the numerators over the common denominator.

$12 - 3 x^{2} - 3 (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}))$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Multiply $- 1$ by $2$ .

$12 - 3 x^{2} - 3 (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}}))$

Step 4.10.2

Subtract $2$ from $3$ .

$12 - 3 x^{2} - 3 (- x^{- 3} (\frac{3}{2} x^{\frac{1}{2}}))$

Step 4.11

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

$12 - 3 x^{2} - 3 (- x^{- 3} \frac{3 x^{\frac{1}{2}}}{2})$

Step 4.12

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

$12 - 3 x^{2} - 3 (- \frac{3 x^{\frac{1}{2}} x^{- 3}}{2})$

Step 4.13

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

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Step 4.13.1

Move $x^{- 3}$ .

$12 - 3 x^{2} - 3 (- \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2})$

Step 4.13.2

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

$12 - 3 x^{2} - 3 (- \frac{3 x^{- 3 + \frac{1}{2}}}{2})$

Step 4.13.3

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

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

Step 4.13.4

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

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

Step 4.13.5

Combine the numerators over the common denominator.

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

Step 4.13.6

Simplify the numerator.

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Step 4.13.6.1

Multiply $- 3$ by $2$ .

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

Step 4.13.6.2

Add $- 6$ and $1$ .

$12 - 3 x^{2} - 3 (- \frac{3 x^{\frac{- 5}{2}}}{2})$

Step 4.13.7

Move the negative in front of the fraction.

$12 - 3 x^{2} - 3 (- \frac{3 x^{- \frac{5}{2}}}{2})$

Step 4.14

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

$12 - 3 x^{2} - 3 (- \frac{3}{2 x^{\frac{5}{2}}})$

Step 4.15

Multiply $- 1$ by $- 3$ .

$12 - 3 x^{2} + 3 \frac{3}{2 x^{\frac{5}{2}}}$

Step 4.16

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

$12 - 3 x^{2} + \frac{3 \cdot 3}{2 x^{\frac{5}{2}}}$

Step 4.17

Multiply $3$ by $3$ .

$12 - 3 x^{2} + \frac{9}{2 x^{\frac{5}{2}}}$

Step 5

Reorder terms.

$- 3 x^{2} + 12 + \frac{9}{2 x^{\frac{5}{2}}}$