Calculus Examples

$\frac{1}{\sqrt{x}}$

Step 1

Find the first derivative.

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Step 1.1

Apply basic rules of exponents.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Move the negative in front of the fraction.

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

Step 1.2

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

$- \frac{1}{2} x^{- \frac{1}{2} - 1}$

Step 1.3

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

$- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 1.4

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

$- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 1.5

Combine the numerators over the common denominator.

$- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2} x^{\frac{- 1 - 2}{2}}$

Step 1.6.2

Subtract $2$ from $- 1$ .

$- \frac{1}{2} x^{\frac{- 3}{2}}$

Step 1.7

Move the negative in front of the fraction.

$- \frac{1}{2} x^{- \frac{3}{2}}$

Step 1.8

Simplify.

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Step 1.8.1

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

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

Step 1.8.2

Combine terms.

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Step 1.8.2.1

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

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

Step 1.8.2.2

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

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

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 $- \frac{1}{2 x^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $- 3$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

Multiply.

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.2

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

$\frac{1}{2} \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}$

Step 2.11

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

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $2$ by $2$ .

$\frac{x^{- \frac{5}{2}} \cdot 3}{4}$

Step 2.12.2

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

$\frac{3 \cdot x^{- \frac{5}{2}}}{4}$

Step 2.12.3

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

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