Calculus Examples

${(x - y - 1)}^{3} = x$ ; $(1, - 1)$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x - y - 1)}^{3}) = \frac{d}{d x} (x)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $x - y - 1$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Replace all occurrences of $u$ with $x - y - 1$ .

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

Step 1.2.2

Differentiate.

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.3

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

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

Step 1.2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 {(x - y - 1)}^{2} (1 - y' + 0)$

Step 1.2.5

Add $1 - y'$ and $0$ .

$3 {(x - y - 1)}^{2} (1 - y')$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Apply the distributive property.

$3 {(x - y - 1)}^{2} \cdot 1 + 3 {(x - y - 1)}^{2} (- y')$

Step 1.2.6.2

Combine terms.

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Step 1.2.6.2.1

Multiply $3$ by $1$ .

$3 {(x - y - 1)}^{2} + 3 {(x - y - 1)}^{2} (- y')$

Step 1.2.6.2.2

Multiply $- 1$ by $3$ .

$3 {(x - y - 1)}^{2} - 3 {(x - y - 1)}^{2} y'$

Step 1.3

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

$1$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Reorder factors in $3 {(x - y - 1)}^{2} - 3 {(x - y - 1)}^{2} y'$ .

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

Step 1.5.2

Subtract $3 {(x - y - 1)}^{2}$ from both sides of the equation.

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

Step 1.5.3

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

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Step 1.5.3.1

Divide each term in $- 3 y' {(x - y - 1)}^{2} = 1 - 3 {(x - y - 1)}^{2}$ by $- 3 {(x - y - 1)}^{2}$ .

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of ${(x - y - 1)}^{2}$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

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

Step 1.5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

To write $\frac{1}{- 3 {(x - y - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 1.5.3.3.2

To write $\frac{- 3 {(x - y - 1)}^{2}}{- 3 {(x - y - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 1.5.3.3.3

Write each expression with a common denominator of $3 {(x - y - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.3.3.1

Multiply $\frac{1}{- 3 {(x - y - 1)}^{2}}$ by $\frac{- 1}{- 1}$ .

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

Step 1.5.3.3.3.2

Multiply $- 1$ by $- 3$ .

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

Step 1.5.3.3.3.3

Multiply $\frac{- 3 {(x - y - 1)}^{2}}{- 3 {(x - y - 1)}^{2}}$ by $\frac{- 1}{- 1}$ .

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

Step 1.5.3.3.3.4

Multiply $- 1$ by $- 3$ .

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

Step 1.5.3.3.4

Combine the numerators over the common denominator.

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

Step 1.5.3.3.5

Multiply $- 1$ by $- 3$ .

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

Step 1.5.3.3.6

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.5.3.3.7

Factor $- 1$ out of $3 {(x - y - 1)}^{2}$ .

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

Step 1.5.3.3.8

Factor $- 1$ out of $- 1 (1) - (- 3 {(x - y - 1)}^{2})$ .

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

Step 1.5.3.3.9

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.3.2

Add $1$ and $1$ .

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

Step 1.7.3.3

Subtract $1$ from $2$ .

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

Step 1.7.3.4

One to any power is one.

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

Step 1.7.3.5

Multiply $- 3$ by $1$ .

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

Step 1.7.3.6

Subtract $3$ from $1$ .

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

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.4.2

Add $1$ and $1$ .

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

Step 1.7.4.3

Subtract $1$ from $2$ .

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

Step 1.7.4.4

One to any power is one.

$- \frac{- 2}{3 \cdot 1}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Multiply $3$ by $1$ .

$- \frac{- 2}{3}$

Step 1.7.5.2

Move the negative in front of the fraction.

$- - \frac{2}{3}$

Step 1.7.6

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

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Step 1.7.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{2}{3})$

Step 1.7.6.2

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

$\frac{2}{3}$

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 $\frac{2}{3}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.3.1.5.1

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

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

Step 2.3.1.5.2

Multiply $2$ by $- 1$ .

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

Step 2.3.1.6

Move the negative in front of the fraction.

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

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

Subtract $1$ from both sides of the equation.

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 1$ by $3$ .

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

Step 2.3.2.5.2

Subtract $3$ from $- 2$ .

$y = \frac{2 x}{3} + \frac{- 5}{3}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{5}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{2}{3} x - \frac{5}{3}$

Step 3