Calculus Examples

$y = 8 \sqrt{x}$

Step 1

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

$y = 8 x^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (8 x^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

$8 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 4.2

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

$8 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 4.3

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

$8 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4.4

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

$8 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.5

Combine the numerators over the common denominator.

$8 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$8 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 4.6.2

Subtract $2$ from $1$ .

$8 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 4.7

Move the negative in front of the fraction.

$8 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 4.8

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

$8 \frac{x^{- \frac{1}{2}}}{2}$

Step 4.9

Combine $8$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{8 x^{- \frac{1}{2}}}{2}$

Step 4.10

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

$\frac{8}{2 x^{\frac{1}{2}}}$

Step 4.11

Factor $2$ out of $8$ .

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

Step 4.12

Cancel the common factors.

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Step 4.12.1

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

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

Step 4.12.2

Cancel the common factor.

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

Step 4.12.3

Rewrite the expression.

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{4}{x^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{4}{x^{\frac{1}{2}}}$