Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $x^{4}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 2.2.2

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

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

Step 2.2.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2.4

Combine $4$ and $\frac{2}{2}$ .

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $4$ by $2$ .

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

Step 2.2.6.2

Add $1$ and $8$ .

$\frac{d}{d x} [3 x^{\frac{9}{2}}] + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{9}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{9}{2}}]$ .

$3 \frac{d}{d x} [x^{\frac{9}{2}}] + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{9}{2}$ .

$3 (\frac{9}{2} x^{\frac{9}{2} - 1}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.5

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

$3 (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.6

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

$3 (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.7

Combine the numerators over the common denominator.

$3 (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $2$ .

$3 (\frac{9}{2} x^{\frac{9 - 2}{2}}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.8.2

Subtract $2$ from $9$ .

$3 (\frac{9}{2} x^{\frac{7}{2}}) + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.9

Combine $\frac{9}{2}$ and $x^{\frac{7}{2}}$ .

$3 \frac{9 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.10

Combine $3$ and $\frac{9 x^{\frac{7}{2}}}{2}$ .

$\frac{3 (9 x^{\frac{7}{2}})}{2} + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 2.11

Multiply $9$ by $3$ .

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{2} \sqrt{x}}]$

Step 3

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

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Step 3.1

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

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{2} x^{\frac{1}{2}}}]$

Step 3.2

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

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Step 3.2.1

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

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{2 + \frac{1}{2}}}]$

Step 3.2.2

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

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{2 \cdot \frac{2}{2} + \frac{1}{2}}}]$

Step 3.2.3

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

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{\frac{2 \cdot 2}{2} + \frac{1}{2}}}]$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{\frac{2 \cdot 2 + 1}{2}}}]$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $2$ by $2$ .

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{\frac{4 + 1}{2}}}]$

Step 3.2.5.2

Add $4$ and $1$ .

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [\frac{- 3}{x^{\frac{5}{2}}}]$

Step 3.3

Move the negative in front of the fraction.

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{\frac{5}{2}}}]$

Step 3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x^{\frac{5}{2}}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$

Step 3.5

Rewrite $\frac{1}{x^{\frac{5}{2}}}$ as ${(x^{\frac{5}{2}})}^{- 1}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 \frac{d}{d x} [{(x^{\frac{5}{2}})}^{- 1}]$

Step 3.6

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 $g (x) = x^{\frac{5}{2}}$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $x^{\frac{5}{2}}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.2

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- u^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3

Replace all occurrences of $u$ with $x^{\frac{5}{2}}$ .

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Step 3.6.3.1

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{2 + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3.2

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{2 \cdot \frac{2}{2} + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3.3

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{\frac{2 \cdot 2}{2} + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3.4

Combine the numerators over the common denominator.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{\frac{2 \cdot 2 + 1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3.5

Simplify the numerator.

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Step 3.6.3.5.1

Multiply $2$ by $2$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{\frac{4 + 1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.6.3.5.2

Add $4$ and $1$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{\frac{5}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- {(x^{\frac{5}{2}})}^{- 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.8

Multiply the exponents in ${(x^{\frac{5}{2}})}^{- 2}$ .

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Step 3.8.1

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{\frac{5}{2} \cdot - 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.8.2

Cancel the common factor of $2$ .

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Step 3.8.2.1

Factor $2$ out of $- 2$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{\frac{5}{2} \cdot (2 (- 1))} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.8.2.2

Cancel the common factor.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{\frac{5}{2} \cdot (2 \cdot - 1)} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.8.2.3

Rewrite the expression.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{5 \cdot - 1} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.8.3

Multiply $5$ by $- 1$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 3.9

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.10

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.11

Combine the numerators over the common denominator.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}))$

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $- 1$ by $2$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 2}{2}}))$

Step 3.12.2

Subtract $2$ from $5$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} (\frac{5}{2} x^{\frac{3}{2}}))$

Step 3.13

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- x^{- 5} \frac{5 x^{\frac{3}{2}}}{2})$

Step 3.14

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{\frac{3}{2}} x^{- 5}}{2})$

Step 3.15

Multiply $x^{\frac{3}{2}}$ by $x^{- 5}$ by adding the exponents.

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Step 3.15.1

Move $x^{- 5}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 (x^{- 5} x^{\frac{3}{2}})}{2})$

Step 3.15.2

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

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{- 5 + \frac{3}{2}}}{2})$

Step 3.15.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{- 5 \cdot \frac{2}{2} + \frac{3}{2}}}{2})$

Step 3.15.4

Combine $- 5$ and $\frac{2}{2}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{\frac{- 5 \cdot 2}{2} + \frac{3}{2}}}{2})$

Step 3.15.5

Combine the numerators over the common denominator.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{\frac{- 5 \cdot 2 + 3}{2}}}{2})$

Step 3.15.6

Simplify the numerator.

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Step 3.15.6.1

Multiply $- 5$ by $2$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{\frac{- 10 + 3}{2}}}{2})$

Step 3.15.6.2

Add $- 10$ and $3$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{\frac{- 7}{2}}}{2})$

Step 3.15.7

Move the negative in front of the fraction.

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5 x^{- \frac{7}{2}}}{2})$

Step 3.16

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{27 x^{\frac{7}{2}}}{2} - 3 (- \frac{5}{2 x^{\frac{7}{2}}})$

Step 3.17

Multiply $- 1$ by $- 3$ .

$\frac{27 x^{\frac{7}{2}}}{2} + 3 \frac{5}{2 x^{\frac{7}{2}}}$

Step 3.18

Combine $3$ and $\frac{5}{2 x^{\frac{7}{2}}}$ .

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{3 \cdot 5}{2 x^{\frac{7}{2}}}$

Step 3.19

Multiply $3$ by $5$ .

$\frac{27 x^{\frac{7}{2}}}{2} + \frac{15}{2 x^{\frac{7}{2}}}$