Calculus Examples

$\frac{d}{d x} (4 \sqrt{x})$

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

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

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

Step 3

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

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

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 $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

Combine $4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 12

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

$\frac{4}{2 x^{\frac{1}{2}}}$

Step 13

Factor $2$ out of $4$ .

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

Step 14

Cancel the common factors.

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Step 14.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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