Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 1.8

Simplify.

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Step 1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8.2

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $- 1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify the expression.

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Step 2.11.1

Multiply $2$ by $2$ .

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

Step 2.11.2

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

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

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 x^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $- 3$ .

$f''' (x) = - \frac{1}{4} \cdot (- \frac{3}{2} x^{\frac{- 5}{2}})$

Step 3.8

Move the negative in front of the fraction.

$f''' (x) = - \frac{1}{4} \cdot (- \frac{3}{2} x^{- \frac{5}{2}})$

Step 3.9

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

$f''' (x) = - \frac{1}{4} \cdot (- \frac{x^{- \frac{5}{2}} \cdot 3}{2})$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{1}{4}) \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}$

Step 3.10.2

Multiply $\frac{1}{4}$ by $1$ .

$f''' (x) = \frac{1}{4} \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}$

Step 3.11

Multiply $\frac{1}{4}$ by $\frac{x^{- \frac{5}{2}} \cdot 3}{2}$ .

$f''' (x) = \frac{x^{- \frac{5}{2}} \cdot 3}{4 \cdot 2}$

Step 3.12

Simplify the expression.

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Step 3.12.1

Multiply $4$ by $2$ .

$f''' (x) = \frac{x^{- \frac{5}{2}} \cdot 3}{8}$

Step 3.12.2

Move $3$ to the left of $x^{- \frac{5}{2}}$ .

$f''' (x) = \frac{3 \cdot x^{- \frac{5}{2}}}{8}$

Step 3.12.3

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

$f''' (x) = \frac{3}{8 x^{\frac{5}{2}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{3}{8}$ is constant with respect to $x$ , the derivative of $\frac{3}{8 x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{3}{8} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{5}{2}}})$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

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

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} ({(x^{\frac{5}{2}})}^{- 1})$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{5}{2}})}^{- 1}$ .

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Step 4.2.2.1

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

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (x^{\frac{5}{2} \cdot - 1})$

Step 4.2.2.2

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 4.2.2.2.1

Combine $\frac{5}{2}$ and $- 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (x^{\frac{5 \cdot - 1}{2}})$

Step 4.2.2.2.2

Multiply $5$ by $- 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (x^{\frac{- 5}{2}})$

Step 4.2.2.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (x^{- \frac{5}{2}})$

Step 4.3

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

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{- \frac{5}{2} - 1})$

Step 4.4

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

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{- \frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 4.5

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

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{- \frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.6

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{\frac{- 5 - 1 \cdot 2}{2}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{\frac{- 5 - 2}{2}})$

Step 4.7.2

Subtract $2$ from $- 5$ .

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{\frac{- 7}{2}})$

Step 4.8

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{5}{2} x^{- \frac{7}{2}})$

Step 4.9

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

$f^{4} (x) = \frac{3}{8} \cdot (- \frac{x^{- \frac{7}{2}} \cdot 5}{2})$

Step 4.10

Multiply $\frac{3}{8}$ by $\frac{x^{- \frac{7}{2}} \cdot 5}{2}$ .

$f^{4} (x) = - \frac{3 (x^{- \frac{7}{2}} \cdot 5)}{8 \cdot 2}$

Step 4.11

Multiply.

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Step 4.11.1

Multiply $5$ by $3$ .

$f^{4} (x) = - \frac{15 x^{- \frac{7}{2}}}{8 \cdot 2}$

Step 4.11.2

Multiply $8$ by $2$ .

$f^{4} (x) = - \frac{15 x^{- \frac{7}{2}}}{16}$

Step 4.11.3

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

$f^{4} (x) = - \frac{15}{16 x^{\frac{7}{2}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{15}{16 x^{\frac{7}{2}}}$ .

$- \frac{15}{16 x^{\frac{7}{2}}}$