Calculus Examples

$\frac{4}{\sqrt{x}}$

Step 1

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

$\frac{d}{d x} [\frac{4}{x^{\frac{1}{2}}}]$

Step 2

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

$4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 4

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

$4 (- \frac{1}{2} x^{- \frac{1}{2} - 1})$

Step 5

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

$4 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$4 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$4 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$4 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}})$

Step 8.2

Subtract $2$ from $- 1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

Multiply $- 1$ by $4$ .

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

Step 12

Combine $- 4$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

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

Step 13

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

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

Step 14

Factor $2$ out of $- 4$ .

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

Step 15

Cancel the common factors.

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Step 15.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

Move the negative in front of the fraction.

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