Calculus Examples

$x^{3} + y^{3} = 22 x y$ ; $(11, 11)$

Step 1

Find the first derivative and evaluate at $x = 11$ and $y = 11$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (22 x y)$

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 $22$ is constant with respect to $x$ , the derivative of $22 x y$ with respect to $x$ is $22 \frac{d}{d x} [x y]$ .

$22 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$22 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 1.3.3

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

$22 (x y' + y \frac{d}{d x} [x])$

Step 1.3.4

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

$22 (x y' + y \cdot 1)$

Step 1.3.5

Multiply $y$ by $1$ .

$22 (x y' + y)$

Step 1.3.6

Apply the distributive property.

$22 x y' + 22 y$

Step 1.4

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

$3 x^{2} + 3 y^{2} y' = 22 x y' + 22 y$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $22 x y'$ from both sides of the equation.

$3 x^{2} + 3 y^{2} y' - 22 x y' = 22 y$

Step 1.5.2

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

$3 y^{2} y' - 22 x y' = 22 y - 3 x^{2}$

Step 1.5.3

Factor $y'$ out of $3 y^{2} y' - 22 x y'$ .

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Step 1.5.3.1

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

$y' (3 y^{2}) - 22 x y' = 22 y - 3 x^{2}$

Step 1.5.3.2

Factor $y'$ out of $- 22 x y'$ .

$y' (3 y^{2}) + y' (- 22 x) = 22 y - 3 x^{2}$

Step 1.5.3.3

Factor $y'$ out of $y' (3 y^{2}) + y' (- 22 x)$ .

$y' (3 y^{2} - 22 x) = 22 y - 3 x^{2}$

Step 1.5.4

Divide each term in $y' (3 y^{2} - 22 x) = 22 y - 3 x^{2}$ by $3 y^{2} - 22 x$ and simplify.

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Step 1.5.4.1

Divide each term in $y' (3 y^{2} - 22 x) = 22 y - 3 x^{2}$ by $3 y^{2} - 22 x$ .

$\frac{y' (3 y^{2} - 22 x)}{3 y^{2} - 22 x} = \frac{22 y}{3 y^{2} - 22 x} + \frac{- 3 x^{2}}{3 y^{2} - 22 x}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $3 y^{2} - 22 x$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{2} - 22 x)}{3 y^{2} - 22 x} = \frac{22 y}{3 y^{2} - 22 x} + \frac{- 3 x^{2}}{3 y^{2} - 22 x}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{22 y}{3 y^{2} - 22 x} + \frac{- 3 x^{2}}{3 y^{2} - 22 x}$

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.

$y' = \frac{22 y - 3 x^{2}}{3 y^{2} - 22 x}$

Step 1.6

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

$\frac{d y}{d x} = \frac{22 y - 3 x^{2}}{3 y^{2} - 22 x}$

Step 1.7

Evaluate at $x = 11$ and $y = 11$ .

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Step 1.7.1

Replace the variable $x$ with $11$ in the expression.

$\frac{22 y - 3 {(11)}^{2}}{3 y^{2} - 22 \cdot 11}$

Step 1.7.2

Replace the variable $y$ with $11$ in the expression.

$\frac{22 (11) - 3 {(11)}^{2}}{3 {(11)}^{2} - 22 \cdot 11}$

Step 1.7.3

Cancel the common factor of $22 (11) - 3 {(11)}^{2}$ and $3 {(11)}^{2} - 22 \cdot 11$ .

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Step 1.7.3.1

Reorder terms.

$\frac{- 3 {(11)}^{2} + 11 \cdot 22}{3 {(11)}^{2} - 22 \cdot 11}$

Step 1.7.3.2

Factor $11$ out of $- 3 {(11)}^{2}$ .

$\frac{11 (- 3 \cdot 11) + 11 \cdot 22}{3 {(11)}^{2} - 22 \cdot 11}$

Step 1.7.3.3

Factor $11$ out of $11 \cdot 22$ .

$\frac{11 (- 3 \cdot 11) + 11 (22)}{3 {(11)}^{2} - 22 \cdot 11}$

Step 1.7.3.4

Factor $11$ out of $11 (- 3 \cdot 11) + 11 (22)$ .

$\frac{11 (- 3 \cdot 11 + 22)}{3 {(11)}^{2} - 22 \cdot 11}$

Step 1.7.3.5

Cancel the common factors.

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Step 1.7.3.5.1

Factor $11$ out of $3 {(11)}^{2}$ .

$\frac{11 (- 3 \cdot 11 + 22)}{11 (3 \cdot 11) - 22 \cdot 11}$

Step 1.7.3.5.2

Factor $11$ out of $- 22 \cdot 11$ .

$\frac{11 (- 3 \cdot 11 + 22)}{11 (3 \cdot 11) + 11 (- 2 \cdot 11)}$

Step 1.7.3.5.3

Factor $11$ out of $11 (3 \cdot 11) + 11 (- 2 \cdot 11)$ .

$\frac{11 (- 3 \cdot 11 + 22)}{11 (3 \cdot 11 - 2 \cdot 11)}$

Step 1.7.3.5.4

Cancel the common factor.

$\frac{11 (- 3 \cdot 11 + 22)}{11 (3 \cdot 11 - 2 \cdot 11)}$

Step 1.7.3.5.5

Rewrite the expression.

$\frac{- 3 \cdot 11 + 22}{3 \cdot 11 - 2 \cdot 11}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Multiply $- 3$ by $11$ .

$\frac{- 33 + 22}{3 \cdot 11 - 2 \cdot 11}$

Step 1.7.4.2

Add $- 33$ and $22$ .

$\frac{- 11}{3 \cdot 11 - 2 \cdot 11}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Multiply $3$ by $11$ .

$\frac{- 11}{33 - 2 \cdot 11}$

Step 1.7.5.2

Multiply $- 2$ by $11$ .

$\frac{- 11}{33 - 22}$

Step 1.7.5.3

Subtract $22$ from $33$ .

$\frac{- 11}{11}$

Step 1.7.6

Divide $- 11$ by $11$ .

$- 1$

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 $(11, 11)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (11) = - 1 \cdot (x - (11))$

Step 2.2

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

$y - 11 = - 1 \cdot (x - 11)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 11)$ .

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Step 2.3.1.1

Rewrite.

$y - 11 = 0 + 0 - 1 \cdot (x - 11)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 11 = - 1 \cdot (x - 11)$

Step 2.3.1.3

Apply the distributive property.

$y - 11 = - 1 x - 1 \cdot - 11$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

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

$y - 11 = - x - 1 \cdot - 11$

Step 2.3.1.4.2

Multiply $- 1$ by $- 11$ .

$y - 11 = - x + 11$

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 $11$ to both sides of the equation.

$y = - x + 11 + 11$

Step 2.3.2.2

Add $11$ and $11$ .

$y = - x + 22$

Step 3