Calculus Examples

$y = \sqrt[4]{x}$ , $(1, 1)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 1$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{4}$ .

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

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 1.4

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}$

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Multiply $- 1$ by $4$ .

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

Step 1.6.2

Subtract $4$ from $1$ .

$\frac{1}{4} x^{\frac{- 3}{4}}$

Step 1.7

Move the negative in front of the fraction.

$\frac{1}{4} x^{- \frac{3}{4}}$

Step 1.8

Simplify.

Tap for more steps...

Step 1.8.1

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

$\frac{1}{4} \cdot \frac{1}{x^{\frac{3}{4}}}$

Step 1.8.2

Multiply $\frac{1}{4}$ by $\frac{1}{x^{\frac{3}{4}}}$ .

$\frac{1}{4 x^{\frac{3}{4}}}$

Step 1.9

Evaluate the derivative at $x = 1$ .

$\frac{1}{4 {(1)}^{\frac{3}{4}}}$

Step 1.10

Simplify.

Tap for more steps...

Step 1.10.1

One to any power is one.

$\frac{1}{4 \cdot 1}$

Step 1.10.2

Multiply $4$ by $1$ .

$\frac{1}{4}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

Tap for more steps...

Step 2.1

Use the slope $\frac{1}{4}$ and a given point $(1, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{1}{4} \cdot (x - (1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 1 = \frac{1}{4} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $\frac{1}{4} \cdot (x - 1)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{1}{4} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = \frac{1}{4} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = \frac{1}{4} x + \frac{1}{4} \cdot - 1$

Step 2.3.1.4

Combine $\frac{1}{4}$ and $x$ .

$y - 1 = \frac{x}{4} + \frac{1}{4} \cdot - 1$

Step 2.3.1.5

Combine $\frac{1}{4}$ and $- 1$ .

$y - 1 = \frac{x}{4} + \frac{- 1}{4}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 1 = \frac{x}{4} - \frac{1}{4}$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = \frac{x}{4} - \frac{1}{4} + 1$

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

$y = \frac{x}{4} - \frac{1}{4} + \frac{4}{4}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{- 1 + 4}{4}$

Step 2.3.2.4

Add $- 1$ and $4$ .

$y = \frac{x}{4} + \frac{3}{4}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{4} x + \frac{3}{4}$

Step 3