Calculus Examples

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

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

Apply basic rules of exponents.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $- 1$ .

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

Step 4.7

Move the negative in front of the fraction.

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

Step 4.8

Simplify.

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Step 4.8.1

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 4.8.2

Combine terms.

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Step 4.8.2.1

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

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

Step 4.8.2.2

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

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

Step 5

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

$y' = - \frac{1}{2 x^{\frac{3}{2}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{1}{2 x^{\frac{3}{2}}}$