Calculus Examples

$x^{3} + y^{3} = 34 x y$ ; $(17, 17)$

Step 1

Find the first derivative and evaluate at $x = 17$ and $y = 17$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (34 x y)$

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 $34$ is constant with respect to $x$ , the derivative of $34 x y$ with respect to $x$ is $34 \frac{d}{d x} [x y]$ .

$34 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$34 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 1.3.3

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

$34 (x y' + y \frac{d}{d x} [x])$

Step 1.3.4

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

$34 (x y' + y \cdot 1)$

Step 1.3.5

Multiply $y$ by $1$ .

$34 (x y' + y)$

Step 1.3.6

Apply the distributive property.

$34 x y' + 34 y$

Step 1.4

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

$3 x^{2} + 3 y^{2} y' = 34 x y' + 34 y$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $34 x y'$ from both sides of the equation.

$3 x^{2} + 3 y^{2} y' - 34 x y' = 34 y$

Step 1.5.2

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

$3 y^{2} y' - 34 x y' = 34 y - 3 x^{2}$

Step 1.5.3

Factor $y'$ out of $3 y^{2} y' - 34 x y'$ .

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Step 1.5.3.1

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

$y' (3 y^{2}) - 34 x y' = 34 y - 3 x^{2}$

Step 1.5.3.2

Factor $y'$ out of $- 34 x y'$ .

$y' (3 y^{2}) + y' (- 34 x) = 34 y - 3 x^{2}$

Step 1.5.3.3

Factor $y'$ out of $y' (3 y^{2}) + y' (- 34 x)$ .

$y' (3 y^{2} - 34 x) = 34 y - 3 x^{2}$

Step 1.5.4

Divide each term in $y' (3 y^{2} - 34 x) = 34 y - 3 x^{2}$ by $3 y^{2} - 34 x$ and simplify.

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Step 1.5.4.1

Divide each term in $y' (3 y^{2} - 34 x) = 34 y - 3 x^{2}$ by $3 y^{2} - 34 x$ .

$\frac{y' (3 y^{2} - 34 x)}{3 y^{2} - 34 x} = \frac{34 y}{3 y^{2} - 34 x} + \frac{- 3 x^{2}}{3 y^{2} - 34 x}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $3 y^{2} - 34 x$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{2} - 34 x)}{3 y^{2} - 34 x} = \frac{34 y}{3 y^{2} - 34 x} + \frac{- 3 x^{2}}{3 y^{2} - 34 x}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{34 y}{3 y^{2} - 34 x} + \frac{- 3 x^{2}}{3 y^{2} - 34 x}$

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.

$y' = \frac{34 y - 3 x^{2}}{3 y^{2} - 34 x}$

Step 1.6

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

$\frac{d y}{d x} = \frac{34 y - 3 x^{2}}{3 y^{2} - 34 x}$

Step 1.7

Evaluate at $x = 17$ and $y = 17$ .

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Step 1.7.1

Replace the variable $x$ with $17$ in the expression.

$\frac{34 y - 3 {(17)}^{2}}{3 y^{2} - 34 \cdot 17}$

Step 1.7.2

Replace the variable $y$ with $17$ in the expression.

$\frac{34 (17) - 3 {(17)}^{2}}{3 {(17)}^{2} - 34 \cdot 17}$

Step 1.7.3

Cancel the common factor of $34 (17) - 3 {(17)}^{2}$ and $3 {(17)}^{2} - 34 \cdot 17$ .

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Step 1.7.3.1

Reorder terms.

$\frac{- 3 {(17)}^{2} + 17 \cdot 34}{3 {(17)}^{2} - 34 \cdot 17}$

Step 1.7.3.2

Factor $17$ out of $- 3 {(17)}^{2}$ .

$\frac{17 (- 3 \cdot 17) + 17 \cdot 34}{3 {(17)}^{2} - 34 \cdot 17}$

Step 1.7.3.3

Factor $17$ out of $17 \cdot 34$ .

$\frac{17 (- 3 \cdot 17) + 17 (34)}{3 {(17)}^{2} - 34 \cdot 17}$

Step 1.7.3.4

Factor $17$ out of $17 (- 3 \cdot 17) + 17 (34)$ .

$\frac{17 (- 3 \cdot 17 + 34)}{3 {(17)}^{2} - 34 \cdot 17}$

Step 1.7.3.5

Cancel the common factors.

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Step 1.7.3.5.1

Factor $17$ out of $3 {(17)}^{2}$ .

$\frac{17 (- 3 \cdot 17 + 34)}{17 (3 \cdot 17) - 34 \cdot 17}$

Step 1.7.3.5.2

Factor $17$ out of $- 34 \cdot 17$ .

$\frac{17 (- 3 \cdot 17 + 34)}{17 (3 \cdot 17) + 17 (- 2 \cdot 17)}$

Step 1.7.3.5.3

Factor $17$ out of $17 (3 \cdot 17) + 17 (- 2 \cdot 17)$ .

$\frac{17 (- 3 \cdot 17 + 34)}{17 (3 \cdot 17 - 2 \cdot 17)}$

Step 1.7.3.5.4

Cancel the common factor.

$\frac{17 (- 3 \cdot 17 + 34)}{17 (3 \cdot 17 - 2 \cdot 17)}$

Step 1.7.3.5.5

Rewrite the expression.

$\frac{- 3 \cdot 17 + 34}{3 \cdot 17 - 2 \cdot 17}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Multiply $- 3$ by $17$ .

$\frac{- 51 + 34}{3 \cdot 17 - 2 \cdot 17}$

Step 1.7.4.2

Add $- 51$ and $34$ .

$\frac{- 17}{3 \cdot 17 - 2 \cdot 17}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Multiply $3$ by $17$ .

$\frac{- 17}{51 - 2 \cdot 17}$

Step 1.7.5.2

Multiply $- 2$ by $17$ .

$\frac{- 17}{51 - 34}$

Step 1.7.5.3

Subtract $34$ from $51$ .

$\frac{- 17}{17}$

Step 1.7.6

Divide $- 17$ by $17$ .

$- 1$

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 $(17, 17)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (17) = - 1 \cdot (x - (17))$

Step 2.2

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

$y - 17 = - 1 \cdot (x - 17)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 17)$ .

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Step 2.3.1.1

Rewrite.

$y - 17 = 0 + 0 - 1 \cdot (x - 17)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 17 = - 1 \cdot (x - 17)$

Step 2.3.1.3

Apply the distributive property.

$y - 17 = - 1 x - 1 \cdot - 17$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

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

$y - 17 = - x - 1 \cdot - 17$

Step 2.3.1.4.2

Multiply $- 1$ by $- 17$ .

$y - 17 = - x + 17$

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 $17$ to both sides of the equation.

$y = - x + 17 + 17$

Step 2.3.2.2

Add $17$ and $17$ .

$y = - x + 34$

Step 3