Calculus Examples

$\sqrt[3]{x} + \sqrt[3]{y^{4}} = 2$ , $(1, 1)$

Step 1

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

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Step 1.1

Rewrite the left side with rational exponents.

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Step 1.1.1

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

$x^{\frac{1}{3}} + \sqrt[3]{y^{4}} = 2$

Step 1.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{4}}$ as $y^{\frac{4}{3}}$ .

$x^{\frac{1}{3}} + y^{\frac{4}{3}} = 2$

Step 1.2

Differentiate both sides of the equation.

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

Step 1.3

Differentiate the left side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $x^{\frac{1}{3}} + y^{\frac{4}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{3}}] + \frac{d}{d x} [y^{\frac{4}{3}}]$ .

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

Step 1.3.2

Evaluate $\frac{d}{d x} [x^{\frac{1}{3}}]$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

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

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

Step 1.3.2.4

Combine the numerators over the common denominator.

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

Step 1.3.2.5

Simplify the numerator.

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Step 1.3.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.3.2.5.2

Subtract $3$ from $1$ .

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

Step 1.3.2.6

Move the negative in front of the fraction.

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

Step 1.3.3

Evaluate $\frac{d}{d x} [y^{\frac{4}{3}}]$ .

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Step 1.3.3.1

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

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Step 1.3.3.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 1.3.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 1.3.3.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 1.3.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{4}{3} - 1} y'$

Step 1.3.3.3

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

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{4}{3} - 1 \cdot \frac{3}{3}} y'$

Step 1.3.3.4

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

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} y'$

Step 1.3.3.5

Combine the numerators over the common denominator.

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{4 - 1 \cdot 3}{3}} y'$

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{4 - 3}{3}} y'$

Step 1.3.3.6.2

Subtract $3$ from $4$ .

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{1}{3}} y'$

Step 1.3.3.7

Combine $\frac{4}{3}$ and $y^{\frac{1}{3}}$ .

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4 y^{\frac{1}{3}}}{3} y'$

Step 1.3.3.8

Combine $\frac{4 y^{\frac{1}{3}}}{3}$ and $y'$ .

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 1.3.4

Simplify.

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Step 1.3.4.1

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

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 1.3.4.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

$\frac{1}{3 x^{\frac{2}{3}}} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0$

Step 1.5

Reform the equation by setting the left side equal to the right side.

$\frac{1}{3 x^{\frac{2}{3}}} + \frac{4 y^{\frac{1}{3}} y'}{3} = 0$

Step 1.6

Solve for $y'$ .

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Step 1.6.1

Reorder factors in $\frac{1}{3 x^{\frac{2}{3}}} + \frac{4 y^{\frac{1}{3}} y'}{3}$ .

$\frac{1}{3 x^{\frac{2}{3}}} + \frac{4 y' y^{\frac{1}{3}}}{3} = 0$

Step 1.6.2

Subtract $\frac{1}{3 x^{\frac{2}{3}}}$ from both sides of the equation.

$\frac{4 y' y^{\frac{1}{3}}}{3} = - \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.6.3

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 y' y^{\frac{1}{3}}}{3} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4

Simplify both sides of the equation.

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Step 1.6.4.1

Simplify the left side.

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Step 1.6.4.1.1

Simplify $\frac{3}{4} \cdot \frac{4 y' y^{\frac{1}{3}}}{3}$ .

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Step 1.6.4.1.1.1

Combine.

$\frac{3 (4 y' y^{\frac{1}{3}})}{4 \cdot 3} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4.1.1.2

Cancel the common factor.

$\frac{3 (4 y' y^{\frac{1}{3}})}{4 \cdot 3} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4.1.1.3

Rewrite the expression.

$\frac{4 y' y^{\frac{1}{3}}}{4} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4.1.1.4

Cancel the common factor.

$\frac{4 y' y^{\frac{1}{3}}}{4} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4.1.1.5

Divide $y' y^{\frac{1}{3}}$ by $1$ .

$y' y^{\frac{1}{3}} = \frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 1.6.4.2

Simplify the right side.

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Step 1.6.4.2.1

Simplify $\frac{3}{4} (- \frac{1}{3 x^{\frac{2}{3}}})$ .

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Step 1.6.4.2.1.1

Cancel the common factor of $3$ .

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Step 1.6.4.2.1.1.1

Move the leading negative in $- \frac{1}{3 x^{\frac{2}{3}}}$ into the numerator.

$y' y^{\frac{1}{3}} = \frac{3}{4} \cdot \frac{- 1}{3 x^{\frac{2}{3}}}$

Step 1.6.4.2.1.1.2

Cancel the common factor.

$y' y^{\frac{1}{3}} = \frac{3}{4} \cdot \frac{- 1}{3 x^{\frac{2}{3}}}$

Step 1.6.4.2.1.1.3

Rewrite the expression.

$y' y^{\frac{1}{3}} = \frac{1}{4} \cdot \frac{- 1}{x^{\frac{2}{3}}}$

Step 1.6.4.2.1.2

Multiply $\frac{1}{4}$ by $\frac{- 1}{x^{\frac{2}{3}}}$ .

$y' y^{\frac{1}{3}} = \frac{- 1}{4 x^{\frac{2}{3}}}$

Step 1.6.4.2.1.3

Move the negative in front of the fraction.

$y' y^{\frac{1}{3}} = - \frac{1}{4 x^{\frac{2}{3}}}$

Step 1.6.5

Divide each term in $y' y^{\frac{1}{3}} = - \frac{1}{4 x^{\frac{2}{3}}}$ by $y^{\frac{1}{3}}$ and simplify.

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Step 1.6.5.1

Divide each term in $y' y^{\frac{1}{3}} = - \frac{1}{4 x^{\frac{2}{3}}}$ by $y^{\frac{1}{3}}$ .

$\frac{y' y^{\frac{1}{3}}}{y^{\frac{1}{3}}} = \frac{- \frac{1}{4 x^{\frac{2}{3}}}}{y^{\frac{1}{3}}}$

Step 1.6.5.2

Simplify the left side.

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Step 1.6.5.2.1

Cancel the common factor.

$\frac{y' y^{\frac{1}{3}}}{y^{\frac{1}{3}}} = \frac{- \frac{1}{4 x^{\frac{2}{3}}}}{y^{\frac{1}{3}}}$

Step 1.6.5.2.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{1}{4 x^{\frac{2}{3}}}}{y^{\frac{1}{3}}}$

Step 1.6.5.3

Simplify the right side.

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Step 1.6.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{1}{4 x^{\frac{2}{3}}} \cdot \frac{1}{y^{\frac{1}{3}}}$

Step 1.6.5.3.2

Multiply $\frac{1}{y^{\frac{1}{3}}}$ by $\frac{1}{4 x^{\frac{2}{3}}}$ .

$y' = - \frac{1}{y^{\frac{1}{3}} (4 x^{\frac{2}{3}})}$

Step 1.6.5.3.3

Move $4$ to the left of $y^{\frac{1}{3}}$ .

$y' = - \frac{1}{4 y^{\frac{1}{3}} x^{\frac{2}{3}}}$

Step 1.7

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{1}{4 y^{\frac{1}{3}} x^{\frac{2}{3}}}$

Step 1.8

Evaluate at $x = 1$ and $y = 1$ .

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Step 1.8.1

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

$- \frac{1}{4 y^{\frac{1}{3}} {(1)}^{\frac{2}{3}}}$

Step 1.8.2

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

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

Step 1.8.3

Multiply ${(1)}^{\frac{1}{3}}$ by ${(1)}^{\frac{2}{3}}$ by adding the exponents.

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Step 1.8.3.1

Move ${(1)}^{\frac{2}{3}}$ .

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

Step 1.8.3.2

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

$- \frac{1}{4 \cdot 1^{\frac{2}{3} + \frac{1}{3}}}$

Step 1.8.3.3

Combine the numerators over the common denominator.

$- \frac{1}{4 \cdot 1^{\frac{2 + 1}{3}}}$

Step 1.8.3.4

Add $2$ and $1$ .

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

Step 1.8.3.5

Divide $3$ by $3$ .

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

Step 1.8.4

Simplify $4 \cdot 1^{1}$ .

$- \frac{1}{4}$

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 $- \frac{1}{4}$ 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 $y$ .

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Step 2.3.1

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

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

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

Step 2.3.1.5

Multiply $- \frac{1}{4} \cdot - 1$ .

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Step 2.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.5.2

Multiply $\frac{1}{4}$ by $1$ .

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

Step 2.3.2

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

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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 $4$ .

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

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{5}{4}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{4} x + \frac{5}{4}$

Step 3