Calculus Examples

$f (x) = 5 \sqrt{x^{5}}$

Step 1

Find the first derivative.

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Step 1.1

Rewrite $\frac{d}{d x} [5 \sqrt{x^{5}}]$ as $\frac{d}{d x} [5 (x^{2} \sqrt{x})]$ .

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Step 1.1.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 1.1.1.1

Factor out $x^{4}$ .

$f' (x) = \frac{d}{d x} (5 \sqrt{x^{4} x})$

Step 1.1.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f' (x) = \frac{d}{d x} (5 \sqrt{{(x^{2})}^{2} x})$

Step 1.1.2

Pull terms out from under the radical.

$f' (x) = \frac{d}{d x} (5 (x^{2} \sqrt{x}))$

Step 1.2

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

$f' (x) = \frac{d}{d x} (5 (x^{2} x^{\frac{1}{2}}))$

Step 1.3

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

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Step 1.3.1

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

$f' (x) = \frac{d}{d x} (5 x^{2 + \frac{1}{2}})$

Step 1.3.2

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

$f' (x) = \frac{d}{d x} (5 x^{2 \cdot \frac{2}{2} + \frac{1}{2}})$

Step 1.3.3

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

$f' (x) = \frac{d}{d x} (5 x^{\frac{2 \cdot 2}{2} + \frac{1}{2}})$

Step 1.3.4

Combine the numerators over the common denominator.

$f' (x) = \frac{d}{d x} (5 x^{\frac{2 \cdot 2 + 1}{2}})$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $2$ by $2$ .

$f' (x) = \frac{d}{d x} (5 x^{\frac{4 + 1}{2}})$

Step 1.3.5.2

Add $4$ and $1$ .

$f' (x) = \frac{d}{d x} (5 x^{\frac{5}{2}})$

Step 1.4

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

$f' (x) = 5 \frac{d}{d x} (x^{\frac{5}{2}})$

Step 1.5

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

$f' (x) = 5 (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 1.6

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

$f' (x) = 5 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 1.7

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

$f' (x) = 5 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.8

Combine the numerators over the common denominator.

$f' (x) = 5 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Multiply $- 1$ by $2$ .

$f' (x) = 5 (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 1.9.2

Subtract $2$ from $5$ .

$f' (x) = 5 (\frac{5}{2} x^{\frac{3}{2}})$

Step 1.10

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

$f' (x) = 5 (\frac{5 x^{\frac{3}{2}}}{2})$

Step 1.11

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

$f' (x) = \frac{5 (5 x^{\frac{3}{2}})}{2}$

Step 1.12

Multiply $5$ by $5$ .

$f' (x) = \frac{25 x^{\frac{3}{2}}}{2}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{25}{2}$ is constant with respect to $x$ , the derivative of $\frac{25 x^{\frac{3}{2}}}{2}$ with respect to $x$ is $\frac{25}{2} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$f'' (x) = \frac{25}{2} \cdot \frac{d}{d x} (x^{\frac{3}{2}})$

Step 2.2

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

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.3

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

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 2.4

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

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 2.6.2

Subtract $2$ from $3$ .

$f'' (x) = \frac{25}{2} \cdot (\frac{3}{2} x^{\frac{1}{2}})$

Step 2.7

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

$f'' (x) = \frac{25}{2} \cdot \frac{3 x^{\frac{1}{2}}}{2}$

Step 2.8

Multiply $\frac{25}{2}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

$f'' (x) = \frac{25 (3 x^{\frac{1}{2}})}{2 \cdot 2}$

Step 2.9

Multiply.

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Step 2.9.1

Multiply $3$ by $25$ .

$f'' (x) = \frac{75 x^{\frac{1}{2}}}{2 \cdot 2}$

Step 2.9.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{75 x^{\frac{1}{2}}}{4}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{75}{4}$ is constant with respect to $x$ , the derivative of $\frac{75 x^{\frac{1}{2}}}{4}$ with respect to $x$ is $\frac{75}{4} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f''' (x) = \frac{75}{4} \cdot \frac{d}{d x} (x^{\frac{1}{2}})$

Step 3.2

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

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.3

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

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.4

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

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.5

Combine the numerators over the common denominator.

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.6.2

Subtract $2$ from $1$ .

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.7

Move the negative in front of the fraction.

$f''' (x) = \frac{75}{4} \cdot (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.8

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

$f''' (x) = \frac{75}{4} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Step 3.9

Multiply $\frac{75}{4}$ by $\frac{x^{- \frac{1}{2}}}{2}$ .

$f''' (x) = \frac{75 x^{- \frac{1}{2}}}{4 \cdot 2}$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $4$ by $2$ .

$f''' (x) = \frac{75 x^{- \frac{1}{2}}}{8}$

Step 3.10.2

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

$f''' (x) = \frac{75}{8 x^{\frac{1}{2}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{75}{8}$ is constant with respect to $x$ , the derivative of $\frac{75}{8 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{75}{8} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$f^{4} (x) = \frac{75}{8} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{1}{2}}})$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

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

$f^{4} (x) = \frac{75}{8} \cdot \frac{d}{d x} ({(x^{\frac{1}{2}})}^{- 1})$

Step 4.2.2

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

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Step 4.2.2.1

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

$f^{4} (x) = \frac{75}{8} \cdot \frac{d}{d x} (x^{\frac{1}{2} \cdot - 1})$

Step 4.2.2.2

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

$f^{4} (x) = \frac{75}{8} \cdot \frac{d}{d x} (x^{\frac{- 1}{2}})$

Step 4.2.2.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{75}{8} \cdot \frac{d}{d x} (x^{- \frac{1}{2}})$

Step 4.3

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

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{- \frac{1}{2} - 1})$

Step 4.4

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

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4.5

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

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.6

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{\frac{- 1 - 2}{2}})$

Step 4.7.2

Subtract $2$ from $- 1$ .

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{\frac{- 3}{2}})$

Step 4.8

Move the negative in front of the fraction.

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{1}{2} x^{- \frac{3}{2}})$

Step 4.9

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

$f^{4} (x) = \frac{75}{8} \cdot (- \frac{x^{- \frac{3}{2}}}{2})$

Step 4.10

Multiply $\frac{75}{8}$ by $\frac{x^{- \frac{3}{2}}}{2}$ .

$f^{4} (x) = - \frac{75 x^{- \frac{3}{2}}}{8 \cdot 2}$

Step 4.11

Multiply.

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Step 4.11.1

Multiply $8$ by $2$ .

$f^{4} (x) = - \frac{75 x^{- \frac{3}{2}}}{16}$

Step 4.11.2

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = - \frac{75}{16 x^{\frac{3}{2}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{75}{16 x^{\frac{3}{2}}}$ .

$- \frac{75}{16 x^{\frac{3}{2}}}$