Calculus Examples

$x^{3} + y^{3} = 4 x y$ ; $(2, 2)$

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

Differentiate both sides of the equation.

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

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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Step 1.2.2.1

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 $g (x) = y$ .

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Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

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

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

Step 1.2.2.1.3

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

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

Step 1.2.2.2

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

$3 x^{2} + 3 y^{2} y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x y$ with respect to $x$ is $4 \frac{d}{d x} [x y]$ .

$4 \frac{d}{d x} [x y]$

Step 1.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$4 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 1.3.3

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

$4 (x y' + y \frac{d}{d x} [x])$

Step 1.3.4

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

$4 (x y' + y \cdot 1)$

Step 1.3.5

Multiply $y$ by $1$ .

$4 (x y' + y)$

Step 1.3.6

Apply the distributive property.

$4 x y' + 4 y$

Step 1.4

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

$3 x^{2} + 3 y^{2} y' = 4 x y' + 4 y$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $4 x y'$ from both sides of the equation.

$3 x^{2} + 3 y^{2} y' - 4 x y' = 4 y$

Step 1.5.2

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

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

Step 1.5.3

Factor $y'$ out of $3 y^{2} y' - 4 x y'$ .

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Step 1.5.3.1

Factor $y'$ out of $3 y^{2} y'$ .

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

Step 1.5.3.2

Factor $y'$ out of $- 4 x y'$ .

$y' (3 y^{2}) + y' (- 4 x) = 4 y - 3 x^{2}$

Step 1.5.3.3

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

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

Step 1.5.4

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

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Step 1.5.4.1

Divide each term in $y' (3 y^{2} - 4 x) = 4 y - 3 x^{2}$ by $3 y^{2} - 4 x$ .

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

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $3 y^{2} - 4 x$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

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

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Combine the numerators over the common denominator.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 2$ and $y = 2$ .

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Step 1.7.1

Replace the variable $x$ with $2$ in the expression.

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

Step 1.7.2

Replace the variable $y$ with $2$ in the expression.

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

Step 1.7.3

Cancel the common factor of $4 (2) - 3 {(2)}^{2}$ and $3 {(2)}^{2} - 4 \cdot 2$ .

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Step 1.7.3.1

Reorder terms.

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

Step 1.7.3.2

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

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

Step 1.7.3.3

Factor $2$ out of $2 \cdot 4$ .

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

Step 1.7.3.4

Factor $2$ out of $2 (- 3 \cdot 2) + 2 (4)$ .

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

Step 1.7.3.5

Cancel the common factors.

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Step 1.7.3.5.1

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

$\frac{2 (- 3 \cdot 2 + 4)}{2 (3 \cdot 2) - 4 \cdot 2}$

Step 1.7.3.5.2

Factor $2$ out of $- 4 \cdot 2$ .

$\frac{2 (- 3 \cdot 2 + 4)}{2 (3 \cdot 2) + 2 (- 2 \cdot 2)}$

Step 1.7.3.5.3

Factor $2$ out of $2 (3 \cdot 2) + 2 (- 2 \cdot 2)$ .

$\frac{2 (- 3 \cdot 2 + 4)}{2 (3 \cdot 2 - 2 \cdot 2)}$

Step 1.7.3.5.4

Cancel the common factor.

$\frac{2 (- 3 \cdot 2 + 4)}{2 (3 \cdot 2 - 2 \cdot 2)}$

Step 1.7.3.5.5

Rewrite the expression.

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

Step 1.7.4

Cancel the common factor of $- 3 \cdot 2 + 4$ and $3 \cdot 2 - 2 \cdot 2$ .

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Step 1.7.4.1

Factor $2$ out of $- 3 \cdot 2$ .

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

Step 1.7.4.2

Factor $2$ out of $4$ .

$\frac{2 \cdot - 3 + 2 (2)}{3 \cdot 2 - 2 \cdot 2}$

Step 1.7.4.3

Factor $2$ out of $2 \cdot - 3 + 2 (2)$ .

$\frac{2 \cdot (- 3 + 2)}{3 \cdot 2 - 2 \cdot 2}$

Step 1.7.4.4

Cancel the common factors.

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Step 1.7.4.4.1

Factor $2$ out of $3 \cdot 2$ .

$\frac{2 \cdot (- 3 + 2)}{2 \cdot 3 - 2 \cdot 2}$

Step 1.7.4.4.2

Factor $2$ out of $- 2 \cdot 2$ .

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

Step 1.7.4.4.3

Factor $2$ out of $2 \cdot 3 + 2 (- 1 \cdot 2)$ .

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

Step 1.7.4.4.4

Cancel the common factor.

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

Step 1.7.4.4.5

Rewrite the expression.

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

Step 1.7.5

Cancel the common factor of $- 3 + 2$ and $3 - 1 \cdot 2$ .

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Step 1.7.5.1

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.7.5.2

Factor $- 1$ out of $- 1 \cdot 2$ .

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

Step 1.7.5.3

Factor $- 1$ out of $- 1 (- 3) - 1 (2)$ .

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

Step 1.7.5.4

Cancel the common factor.

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

Step 1.7.5.5

Rewrite the expression.

$\frac{1}{- 1}$

Step 1.7.5.6

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$- 1 \cdot 1$

Step 1.7.6

Multiply $- 1$ by $1$ .

$- 1$

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

$y - (2) = - 1 \cdot (x - (2))$

Step 2.2

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

$y - 2 = - 1 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - 1 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = - 1 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = - 1 x - 1 \cdot - 2$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

Rewrite $- 1 x$ as $- x$ .

$y - 2 = - x - 1 \cdot - 2$

Step 2.3.1.4.2

Multiply $- 1$ by $- 2$ .

$y - 2 = - x + 2$

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.

$y = - x + 2 + 2$

Step 2.3.2.2

Add $2$ and $2$ .

$y = - x + 4$

Step 3