Calculus Examples

$f (x) = \frac{5}{\sqrt[3]{x^{2}}} - x$ ; , $(- 1, 6)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 6$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $\frac{5}{\sqrt[3]{x^{2}}} - x$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{\sqrt[3]{x^{2}}}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [\frac{5}{\sqrt[3]{x^{2}}}] + \frac{d}{d x} [- x]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{5}{\sqrt[3]{x^{2}}}]$ .

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Step 1.2.1

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

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

Step 1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{\frac{2}{3}}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

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

Step 1.2.3

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

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

Step 1.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 1.2.6

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 2}$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 1.2.6.2.1

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

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

Step 1.2.6.2.2

Multiply $2$ by $- 2$ .

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

Step 1.2.6.3

Move the negative in front of the fraction.

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

Combine the numerators over the common denominator.

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

Step 1.2.10

Simplify the numerator.

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Step 1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 1.2.10.2

Subtract $3$ from $2$ .

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

Step 1.2.11

Move the negative in front of the fraction.

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

Step 1.2.12

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

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

Step 1.2.13

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

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

Step 1.2.14

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 1.2.14.1

Move $x^{- \frac{4}{3}}$ .

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

Step 1.2.14.2

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

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

Step 1.2.14.3

Combine the numerators over the common denominator.

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

Step 1.2.14.4

Subtract $1$ from $- 4$ .

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

Step 1.2.14.5

Move the negative in front of the fraction.

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

Step 1.2.15

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2.16

Multiply $- 1$ by $5$ .

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

Step 1.2.17

Combine $- 5$ and $\frac{2}{3 x^{\frac{5}{3}}}$ .

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

Step 1.2.18

Multiply $- 5$ by $2$ .

$\frac{- 10}{3 x^{\frac{5}{3}}} + \frac{d}{d x} [- x]$

Step 1.2.19

Move the negative in front of the fraction.

$- \frac{10}{3 x^{\frac{5}{3}}} + \frac{d}{d x} [- x]$

Step 1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{10}{3 x^{\frac{5}{3}}} - \frac{d}{d x} [x]$

Step 1.3.2

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

$- \frac{10}{3 x^{\frac{5}{3}}} - 1 \cdot 1$

Step 1.3.3

Multiply $- 1$ by $1$ .

$- \frac{10}{3 x^{\frac{5}{3}}} - 1$

Step 1.4

Reorder terms.

$- 1 - \frac{10}{3 x^{\frac{5}{3}}}$

Step 1.5

Evaluate the derivative at $x = - 1$ .

$- 1 - \frac{10}{3 {(- 1)}^{\frac{5}{3}}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Simplify the denominator.

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Step 1.6.1.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- 1 - \frac{10}{3 {({(- 1)}^{3})}^{\frac{5}{3}}}$

Step 1.6.1.1.2

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

$- 1 - \frac{10}{3 {(- 1)}^{3 (\frac{5}{3})}}$

Step 1.6.1.1.3

Cancel the common factor of $3$ .

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Step 1.6.1.1.3.1

Cancel the common factor.

$- 1 - \frac{10}{3 {(- 1)}^{3 (\frac{5}{3})}}$

Step 1.6.1.1.3.2

Rewrite the expression.

$- 1 - \frac{10}{3 {(- 1)}^{5}}$

Step 1.6.1.1.4

Raise $- 1$ to the power of $5$ .

$- 1 - \frac{10}{3 \cdot - 1}$

Step 1.6.1.2

Multiply $3$ by $- 1$ .

$- 1 - \frac{10}{- 3}$

Step 1.6.1.3

Move the negative in front of the fraction.

$- 1 - - \frac{10}{3}$

Step 1.6.1.4

Multiply $- - \frac{10}{3}$ .

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$- 1 + 1 (\frac{10}{3})$

Step 1.6.1.4.2

Multiply $\frac{10}{3}$ by $1$ .

$- 1 + \frac{10}{3}$

Step 1.6.2

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

$- 1 \cdot \frac{3}{3} + \frac{10}{3}$

Step 1.6.3

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

$\frac{- 1 \cdot 3}{3} + \frac{10}{3}$

Step 1.6.4

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 3 + 10}{3}$

Step 1.6.5

Simplify the numerator.

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Step 1.6.5.1

Multiply $- 1$ by $3$ .

$\frac{- 3 + 10}{3}$

Step 1.6.5.2

Add $- 3$ and $10$ .

$\frac{7}{3}$

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 $\frac{7}{3}$ and a given point $(- 1, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{7}{3} \cdot (x - (- 1))$

Step 2.2

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

$y - 6 = \frac{7}{3} \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{7}{3} \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 + \frac{7}{3} \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = \frac{7}{3} \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = \frac{7}{3} x + \frac{7}{3} \cdot 1$

Step 2.3.1.4

Combine $\frac{7}{3}$ and $x$ .

$y - 6 = \frac{7 x}{3} + \frac{7}{3} \cdot 1$

Step 2.3.1.5

Multiply $\frac{7}{3}$ by $1$ .

$y - 6 = \frac{7 x}{3} + \frac{7}{3}$

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 $6$ to both sides of the equation.

$y = \frac{7 x}{3} + \frac{7}{3} + 6$

Step 2.3.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{7 x}{3} + \frac{7}{3} + 6 \cdot \frac{3}{3}$

Step 2.3.2.3

Combine $6$ and $\frac{3}{3}$ .

$y = \frac{7 x}{3} + \frac{7}{3} + \frac{6 \cdot 3}{3}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{3} + \frac{7 + 6 \cdot 3}{3}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $6$ by $3$ .

$y = \frac{7 x}{3} + \frac{7 + 18}{3}$

Step 2.3.2.5.2

Add $7$ and $18$ .

$y = \frac{7 x}{3} + \frac{25}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{7}{3} x + \frac{25}{3}$

Step 3