Calculus Examples

$y = \sqrt{x}$ , $(9, 3)$

Step 1

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

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

Step 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 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

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 9

Evaluate the derivative at $x = 9$ .

$\frac{1}{2 {(9)}^{\frac{1}{2}}}$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify the denominator.

Tap for more steps...

Step 10.1.1

Rewrite $9$ as $3^{2}$ .

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

Step 10.1.2

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

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

Step 10.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1.3.1

Cancel the common factor.

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

Step 10.1.3.2

Rewrite the expression.

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

Step 10.1.4

Evaluate the exponent.

$\frac{1}{2 \cdot 3}$

Step 10.2

Multiply $2$ by $3$ .

$\frac{1}{6}$