Calculus Examples

$y = \sqrt[5]{2 x^{3} + 8 x}$ , $(2, 2)$

Step 1

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

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 x^{3} + 8 x}$ as ${(2 x^{3} + 8 x)}^{\frac{1}{5}}$ .

$\frac{d}{d x} [{(2 x^{3} + 8 x)}^{\frac{1}{5}}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = 2 x^{3} + 8 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 8 x$ .

$\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

$\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.2.3

Replace all occurrences of $u$ with $2 x^{3} + 8 x$ .

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{1}{5} - 1} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{1}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{1 - 1 \cdot 5}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{1 - 5}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.6.2

Subtract $5$ from $1$ .

$\frac{1}{5} {(2 x^{3} + 8 x)}^{\frac{- 4}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{5} {(2 x^{3} + 8 x)}^{- \frac{4}{5}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.7.2

Combine $\frac{1}{5}$ and ${(2 x^{3} + 8 x)}^{- \frac{4}{5}}$ .

$\frac{{(2 x^{3} + 8 x)}^{- \frac{4}{5}}}{5} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.7.3

Move ${(2 x^{3} + 8 x)}^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} \frac{d}{d x} [2 x^{3} + 8 x]$

Step 1.8

By the Sum Rule, the derivative of $2 x^{3} + 8 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [8 x]$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [8 x])$

Step 1.9

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 x])$

Step 1.10

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

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (2 (3 x^{2}) + \frac{d}{d x} [8 x])$

Step 1.11

Multiply $3$ by $2$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (6 x^{2} + \frac{d}{d x} [8 x])$

Step 1.12

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (6 x^{2} + 8 \frac{d}{d x} [x])$

Step 1.13

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

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (6 x^{2} + 8 \cdot 1)$

Step 1.14

Multiply $8$ by $1$ .

$\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (6 x^{2} + 8)$

Step 1.15

Simplify.

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Step 1.15.1

Reorder the factors of $\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}} (6 x^{2} + 8)$ .

$(6 x^{2} + 8) \frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$

Step 1.15.2

Multiply $6 x^{2} + 8$ by $\frac{1}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$ .

$\frac{6 x^{2} + 8}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$

Step 1.15.3

Factor $2$ out of $6 x^{2} + 8$ .

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Step 1.15.3.1

Factor $2$ out of $6 x^{2}$ .

$\frac{2 (3 x^{2}) + 8}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$

Step 1.15.3.2

Factor $2$ out of $8$ .

$\frac{2 (3 x^{2}) + 2 (4)}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$

Step 1.15.3.3

Factor $2$ out of $2 (3 x^{2}) + 2 (4)$ .

$\frac{2 (3 x^{2} + 4)}{5 {(2 x^{3} + 8 x)}^{\frac{4}{5}}}$

Step 1.16

Evaluate the derivative at $x = 2$ .

$\frac{2 (3 {(2)}^{2} + 4)}{5 {(2 {(2)}^{3} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17

Simplify.

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Step 1.17.1

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

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Step 1.17.1.1

Multiply $2$ by ${(2)}^{3}$ .

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Step 1.17.1.1.1

Raise $2$ to the power of $1$ .

$\frac{2 (3 {(2)}^{2} + 4)}{5 {(2^{1} {(2)}^{3} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (3 {(2)}^{2} + 4)}{5 {(2^{1 + 3} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.1.2

Add $1$ and $3$ .

$\frac{2 (3 {(2)}^{2} + 4)}{5 {(2^{4} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.2

Simplify the numerator.

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Step 1.17.2.1

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

$\frac{2 (3 \cdot 4 + 4)}{5 {(2^{4} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.2.2

Multiply $3$ by $4$ .

$\frac{2 (12 + 4)}{5 {(2^{4} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.2.3

Add $12$ and $4$ .

$\frac{2 \cdot 16}{5 {(2^{4} + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.3

Simplify the denominator.

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Step 1.17.3.1

Simplify each term.

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Step 1.17.3.1.1

Raise $2$ to the power of $4$ .

$\frac{2 \cdot 16}{5 {(16 + 8 (2))}^{\frac{4}{5}}}$

Step 1.17.3.1.2

Multiply $8$ by $2$ .

$\frac{2 \cdot 16}{5 {(16 + 16)}^{\frac{4}{5}}}$

Step 1.17.3.2

Add $16$ and $16$ .

$\frac{2 \cdot 16}{5 \cdot 32^{\frac{4}{5}}}$

Step 1.17.3.3

Rewrite $32$ as $2^{5}$ .

$\frac{2 \cdot 16}{5 \cdot {(2^{5})}^{\frac{4}{5}}}$

Step 1.17.3.4

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

$\frac{2 \cdot 16}{5 \cdot 2^{5 (\frac{4}{5})}}$

Step 1.17.3.5

Cancel the common factor of $5$ .

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Step 1.17.3.5.1

Cancel the common factor.

$\frac{2 \cdot 16}{5 \cdot 2^{5 (\frac{4}{5})}}$

Step 1.17.3.5.2

Rewrite the expression.

$\frac{2 \cdot 16}{5 \cdot 2^{4}}$

Step 1.17.3.6

Raise $2$ to the power of $4$ .

$\frac{2 \cdot 16}{5 \cdot 16}$

Step 1.17.4

Reduce the expression by cancelling the common factors.

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Step 1.17.4.1

Multiply $2$ by $16$ .

$\frac{32}{5 \cdot 16}$

Step 1.17.4.2

Multiply $5$ by $16$ .

$\frac{32}{80}$

Step 1.17.4.3

Cancel the common factor of $32$ and $80$ .

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Step 1.17.4.3.1

Factor $16$ out of $32$ .

$\frac{16 (2)}{80}$

Step 1.17.4.3.2

Cancel the common factors.

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Step 1.17.4.3.2.1

Factor $16$ out of $80$ .

$\frac{16 \cdot 2}{16 \cdot 5}$

Step 1.17.4.3.2.2

Cancel the common factor.

$\frac{16 \cdot 2}{16 \cdot 5}$

Step 1.17.4.3.2.3

Rewrite the expression.

$\frac{2}{5}$

Step 2

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

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Step 2.1

Use the slope $\frac{2}{5}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{2}{5} \cdot (x - (2))$

Step 2.2

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

$y - 2 = \frac{2}{5} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{2}{5} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{2}{5} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = \frac{2}{5} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = \frac{2}{5} x + \frac{2}{5} \cdot - 2$

Step 2.3.1.4

Combine $\frac{2}{5}$ and $x$ .

$y - 2 = \frac{2 x}{5} + \frac{2}{5} \cdot - 2$

Step 2.3.1.5

Multiply $\frac{2}{5} \cdot - 2$ .

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Step 2.3.1.5.1

Combine $\frac{2}{5}$ and $- 2$ .

$y - 2 = \frac{2 x}{5} + \frac{2 \cdot - 2}{5}$

Step 2.3.1.5.2

Multiply $2$ by $- 2$ .

$y - 2 = \frac{2 x}{5} + \frac{- 4}{5}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 2 = \frac{2 x}{5} - \frac{4}{5}$

Step 2.3.2

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

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$y = \frac{2 x}{5} - \frac{4}{5} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{2 x}{5} - \frac{4}{5} + 2 \cdot \frac{5}{5}$

Step 2.3.2.3

Combine $2$ and $\frac{5}{5}$ .

$y = \frac{2 x}{5} - \frac{4}{5} + \frac{2 \cdot 5}{5}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{2 x}{5} + \frac{- 4 + 2 \cdot 5}{5}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $5$ .

$y = \frac{2 x}{5} + \frac{- 4 + 10}{5}$

Step 2.3.2.5.2

Add $- 4$ and $10$ .

$y = \frac{2 x}{5} + \frac{6}{5}$

Step 2.3.3

Reorder terms.

$y = \frac{2}{5} x + \frac{6}{5}$

Step 3