Calculus Examples

$y = \sqrt[3]{x^{2} + 4}$ at $x = 2$

Step 1

Find the corresponding $y$ -value to $x = 2$ .

Tap for more steps...

Step 1.1

Substitute $2$ in for $x$ .

$y = \sqrt[3]{{(2)}^{2} + 4}$

Step 1.2

Simplify $\sqrt[3]{{(2)}^{2} + 4}$ .

Tap for more steps...

Step 1.2.1

Raise $2$ to the power of $2$ .

$y = \sqrt[3]{4 + 4}$

Step 1.2.2

Add $4$ and $4$ .

$y = \sqrt[3]{8}$

Step 1.2.3

Rewrite $8$ as $2^{3}$ .

$y = \sqrt[3]{2^{3}}$

Step 1.2.4

Pull terms out from under the radical, assuming real numbers.

$y = 2$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 2$ to find the slope of the tangent line.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} + 4}$ as ${(x^{2} + 4)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(x^{2} + 4)}^{\frac{1}{3}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x^{2} + 4$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 4$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x^{2} + 4]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x^{2} + 4]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 4$ .

$\frac{1}{3} {(x^{2} + 4)}^{\frac{1}{3} - 1} \frac{d}{d x} [x^{2} + 4]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} {(x^{2} + 4)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{3} {(x^{2} + 4)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{3} {(x^{2} + 4)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(x^{2} + 4)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(x^{2} + 4)}^{\frac{- 2}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.7

Combine fractions.

Tap for more steps...

Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(x^{2} + 4)}^{- \frac{2}{3}} \frac{d}{d x} [x^{2} + 4]$

Step 2.7.2

Combine $\frac{1}{3}$ and ${(x^{2} + 4)}^{- \frac{2}{3}}$ .

$\frac{{(x^{2} + 4)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x^{2} + 4]$

Step 2.7.3

Move ${(x^{2} + 4)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}} \frac{d}{d x} [x^{2} + 4]$

Step 2.8

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])$

Step 2.9

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

$\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}} (2 x + \frac{d}{d x} [4])$

Step 2.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}} (2 x + 0)$

Step 2.11

Combine fractions.

Tap for more steps...

Step 2.11.1

Add $2 x$ and $0$ .

$\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}} (2 x)$

Step 2.11.2

Combine $2$ and $\frac{1}{3 {(x^{2} + 4)}^{\frac{2}{3}}}$ .

$\frac{2}{3 {(x^{2} + 4)}^{\frac{2}{3}}} x$

Step 2.11.3

Combine $\frac{2}{3 {(x^{2} + 4)}^{\frac{2}{3}}}$ and $x$ .

$\frac{2 x}{3 {(x^{2} + 4)}^{\frac{2}{3}}}$

Step 2.12

Evaluate the derivative at $x = 2$ .

$\frac{2 (2)}{3 {({(2)}^{2} + 4)}^{\frac{2}{3}}}$

Step 2.13

Simplify.

Tap for more steps...

Step 2.13.1

Multiply $2$ by $2$ .

$\frac{4}{3 {(2^{2} + 4)}^{\frac{2}{3}}}$

Step 2.13.2

Simplify the denominator.

Tap for more steps...

Step 2.13.2.1

Raise $2$ to the power of $2$ .

$\frac{4}{3 {(4 + 4)}^{\frac{2}{3}}}$

Step 2.13.2.2

Add $4$ and $4$ .

$\frac{4}{3 \cdot 8^{\frac{2}{3}}}$

Step 2.13.2.3

Rewrite $8$ as $2^{3}$ .

$\frac{4}{3 \cdot {(2^{3})}^{\frac{2}{3}}}$

Step 2.13.2.4

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

$\frac{4}{3 \cdot 2^{3 (\frac{2}{3})}}$

Step 2.13.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.13.2.5.1

Cancel the common factor.

$\frac{4}{3 \cdot 2^{3 (\frac{2}{3})}}$

Step 2.13.2.5.2

Rewrite the expression.

$\frac{4}{3 \cdot 2^{2}}$

Step 2.13.2.6

Raise $2$ to the power of $2$ .

$\frac{4}{3 \cdot 4}$

Step 2.13.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.13.3.1

Multiply $3$ by $4$ .

$\frac{4}{12}$

Step 2.13.3.2

Cancel the common factor of $4$ and $12$ .

Tap for more steps...

Step 2.13.3.2.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{12}$

Step 2.13.3.2.2

Cancel the common factors.

Tap for more steps...

Step 2.13.3.2.2.1

Factor $4$ out of $12$ .

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

Step 2.13.3.2.2.2

Cancel the common factor.

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

Step 2.13.3.2.2.3

Rewrite the expression.

$\frac{1}{3}$

Step 3

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

Tap for more steps...

Step 3.1

Use the slope $\frac{1}{3}$ and a given point $(2, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{1}{3} \cdot (x - (2))$

Step 3.2

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

$y - 2 = \frac{1}{3} \cdot (x - 2)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $\frac{1}{3} \cdot (x - 2)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{3} \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 2 = \frac{1}{3} \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 2 = \frac{1}{3} x + \frac{1}{3} \cdot - 2$

Step 3.3.1.4

Combine $\frac{1}{3}$ and $x$ .

$y - 2 = \frac{x}{3} + \frac{1}{3} \cdot - 2$

Step 3.3.1.5

Combine $\frac{1}{3}$ and $- 2$ .

$y - 2 = \frac{x}{3} + \frac{- 2}{3}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - 2 = \frac{x}{3} - \frac{2}{3}$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $2$ to both sides of the equation.

$y = \frac{x}{3} - \frac{2}{3} + 2$

Step 3.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x}{3} - \frac{2}{3} + 2 \cdot \frac{3}{3}$

Step 3.3.2.3

Combine $2$ and $\frac{3}{3}$ .

$y = \frac{x}{3} - \frac{2}{3} + \frac{2 \cdot 3}{3}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{3} + \frac{- 2 + 2 \cdot 3}{3}$

Step 3.3.2.5

Simplify the numerator.

Tap for more steps...

Step 3.3.2.5.1

Multiply $2$ by $3$ .

$y = \frac{x}{3} + \frac{- 2 + 6}{3}$

Step 3.3.2.5.2

Add $- 2$ and $6$ .

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

Step 3.3.3

Reorder terms.

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

Step 4