Calculus Examples

$f (x) = 7 x^{5} \sqrt{x} - \frac{3}{x^{3} \sqrt{x}}$

Step 1

By the Sum Rule, the derivative of $7 x^{5} \sqrt{x} - \frac{3}{x^{3} \sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [7 x^{5} \sqrt{x}] + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [7 x^{5} \sqrt{x}]$ .

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Step 2.1

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

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

Step 2.2

Multiply $x^{5}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $5$ by $2$ .

$\frac{d}{d x} [7 x^{\frac{1 + 10}{2}}] + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.2.6.2

Add $1$ and $10$ .

$\frac{d}{d x} [7 x^{\frac{11}{2}}] + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{\frac{11}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x^{\frac{11}{2}}]$ .

$7 \frac{d}{d x} [x^{\frac{11}{2}}] + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{11}{2}$ .

$7 (\frac{11}{2} x^{\frac{11}{2} - 1}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.5

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

$7 (\frac{11}{2} x^{\frac{11}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.6

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

$7 (\frac{11}{2} x^{\frac{11}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.7

Combine the numerators over the common denominator.

$7 (\frac{11}{2} x^{\frac{11 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $2$ .

$7 (\frac{11}{2} x^{\frac{11 - 2}{2}}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.8.2

Subtract $2$ from $11$ .

$7 (\frac{11}{2} x^{\frac{9}{2}}) + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.9

Combine $\frac{11}{2}$ and $x^{\frac{9}{2}}$ .

$7 \frac{11 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.10

Combine $7$ and $\frac{11 x^{\frac{9}{2}}}{2}$ .

$\frac{7 (11 x^{\frac{9}{2}})}{2} + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 2.11

Multiply $11$ by $7$ .

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{3} \sqrt{x}}]$

Step 3

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

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Step 3.1

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

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{3} x^{\frac{1}{2}}}]$

Step 3.2

Multiply $x^{3}$ 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.

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{3 + \frac{1}{2}}}]$

Step 3.2.2

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

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{3 \cdot \frac{2}{2} + \frac{1}{2}}}]$

Step 3.2.3

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

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{\frac{3 \cdot 2}{2} + \frac{1}{2}}}]$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{\frac{3 \cdot 2 + 1}{2}}}]$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $3$ by $2$ .

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{\frac{6 + 1}{2}}}]$

Step 3.2.5.2

Add $6$ and $1$ .

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{d}{d x} [- \frac{3}{x^{\frac{7}{2}}}]$

Step 3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x^{\frac{7}{2}}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x^{\frac{7}{2}}}]$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 \frac{d}{d x} [\frac{1}{x^{\frac{7}{2}}}]$

Step 3.4

Rewrite $\frac{1}{x^{\frac{7}{2}}}$ as ${(x^{\frac{7}{2}})}^{- 1}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 \frac{d}{d x} [{(x^{\frac{7}{2}})}^{- 1}]$

Step 3.5

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 $g (x) = x^{\frac{7}{2}}$ .

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Step 3.5.1

To apply the Chain Rule, set $u$ as $x^{\frac{7}{2}}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.2

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- u^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3

Replace all occurrences of $u$ with $x^{\frac{7}{2}}$ .

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Step 3.5.3.1

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{3 + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3.2

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{3 \cdot \frac{2}{2} + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3.3

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{\frac{3 \cdot 2}{2} + \frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3.4

Combine the numerators over the common denominator.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{\frac{3 \cdot 2 + 1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3.5

Simplify the numerator.

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Step 3.5.3.5.1

Multiply $3$ by $2$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{\frac{6 + 1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.5.3.5.2

Add $6$ and $1$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{\frac{7}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{2}}])$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{2}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- {(x^{\frac{7}{2}})}^{- 2} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.7

Multiply the exponents in ${(x^{\frac{7}{2}})}^{- 2}$ .

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Step 3.7.1

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{\frac{7}{2} \cdot - 2} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.7.2

Cancel the common factor of $2$ .

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Step 3.7.2.1

Factor $2$ out of $- 2$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{\frac{7}{2} \cdot (2 (- 1))} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.7.2.2

Cancel the common factor.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{\frac{7}{2} \cdot (2 \cdot - 1)} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.7.2.3

Rewrite the expression.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{7 \cdot - 1} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.7.3

Multiply $7$ by $- 1$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{7}{2} - 1}))$

Step 3.8

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.9

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.10

Combine the numerators over the common denominator.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}))$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{7 - 2}{2}}))$

Step 3.11.2

Subtract $2$ from $7$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} (\frac{7}{2} x^{\frac{5}{2}}))$

Step 3.12

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- x^{- 7} \frac{7 x^{\frac{5}{2}}}{2})$

Step 3.13

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{\frac{5}{2}} x^{- 7}}{2})$

Step 3.14

Multiply $x^{\frac{5}{2}}$ by $x^{- 7}$ by adding the exponents.

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Step 3.14.1

Move $x^{- 7}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 (x^{- 7} x^{\frac{5}{2}})}{2})$

Step 3.14.2

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

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{- 7 + \frac{5}{2}}}{2})$

Step 3.14.3

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{- 7 \cdot \frac{2}{2} + \frac{5}{2}}}{2})$

Step 3.14.4

Combine $- 7$ and $\frac{2}{2}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{\frac{- 7 \cdot 2}{2} + \frac{5}{2}}}{2})$

Step 3.14.5

Combine the numerators over the common denominator.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{\frac{- 7 \cdot 2 + 5}{2}}}{2})$

Step 3.14.6

Simplify the numerator.

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Step 3.14.6.1

Multiply $- 7$ by $2$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{\frac{- 14 + 5}{2}}}{2})$

Step 3.14.6.2

Add $- 14$ and $5$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{\frac{- 9}{2}}}{2})$

Step 3.14.7

Move the negative in front of the fraction.

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7 x^{- \frac{9}{2}}}{2})$

Step 3.15

Move $x^{- \frac{9}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{77 x^{\frac{9}{2}}}{2} - 3 (- \frac{7}{2 x^{\frac{9}{2}}})$

Step 3.16

Multiply $- 1$ by $- 3$ .

$\frac{77 x^{\frac{9}{2}}}{2} + 3 \frac{7}{2 x^{\frac{9}{2}}}$

Step 3.17

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

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{3 \cdot 7}{2 x^{\frac{9}{2}}}$

Step 3.18

Multiply $3$ by $7$ .

$\frac{77 x^{\frac{9}{2}}}{2} + \frac{21}{2 x^{\frac{9}{2}}}$