Calculus Examples

$\sqrt{x}$ , $(1, 1)$

Step 1

Find the derivative of the function. To find the slope of the equation tangent to the line, evaluate the derivative at the desired value of $x$ .

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

Step 2

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 3

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

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 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Simplify.

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Step 9.1

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 9.2

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

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

Step 10

The derivative of the equation in terms of $y$ can also be represented as $f' (x)$ .

$f' (x) = \frac{1}{2 x^{\frac{1}{2}}}$

Step 11

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{1}{2 {(1)}^{\frac{1}{2}}}$

Step 12

One to any power is one.

$f' (1) = \frac{1}{2 \cdot 1}$

Step 13

Multiply $2$ by $1$ .

$f' (1) = \frac{1}{2}$

Step 14

The derivative at $(1, 1)$ is $\frac{1}{2}$ .

$\frac{1}{2}$