Calculus Examples

$4 x^{2} + 9 y^{2} = 72$ , $(3, 2)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 x^{2} + 9 y^{2}) = \frac{d}{d x} (72)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{2} + 9 y^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9 y^{2}]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9 y^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 y^{2}]$

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 $n = 2$ .

$4 (2 x) + \frac{d}{d x} [9 y^{2}]$

Step 1.2.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [9 y^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [9 y^{2}]$ .

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Step 1.2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 y^{2}$ with respect to $x$ is $9 \frac{d}{d x} [y^{2}]$ .

$8 x + 9 \frac{d}{d x} [y^{2}]$

Step 1.2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.3.2.1

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

$8 x + 9 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 1.2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$8 x + 9 (2 u \frac{d}{d x} [y])$

Step 1.2.3.2.3

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

$8 x + 9 (2 y \frac{d}{d x} [y])$

Step 1.2.3.3

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

$8 x + 9 (2 y y')$

Step 1.2.3.4

Multiply $2$ by $9$ .

$8 x + 18 y y'$

Step 1.2.4

Reorder terms.

$18 y y' + 8 x$

Step 1.3

Since $72$ is constant with respect to $x$ , the derivative of $72$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$18 y y' + 8 x = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $8 x$ from both sides of the equation.

$18 y y' = - 8 x$

Step 1.5.2

Divide each term in $18 y y' = - 8 x$ by $18 y$ and simplify.

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Step 1.5.2.1

Divide each term in $18 y y' = - 8 x$ by $18 y$ .

$\frac{18 y y'}{18 y} = \frac{- 8 x}{18 y}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $18$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{18 y y'}{18 y} = \frac{- 8 x}{18 y}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 8 x}{18 y}$

Step 1.5.2.2.2

Cancel the common factor of $y$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 8 x}{18 y}$

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 8 x}{18 y}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Cancel the common factor of $- 8$ and $18$ .

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Step 1.5.2.3.1.1

Factor $2$ out of $- 8 x$ .

$y' = \frac{2 (- 4 x)}{18 y}$

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

Factor $2$ out of $18 y$ .

$y' = \frac{2 (- 4 x)}{2 (9 y)}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 4 x)}{2 (9 y)}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{- 4 x}{9 y}$

Step 1.5.2.3.2

Move the negative in front of the fraction.

$y' = - \frac{4 x}{9 y}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{4 x}{9 y}$

Step 1.7

Evaluate at $x = 3$ and $y = 2$ .

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Step 1.7.1

Replace the variable $x$ with $3$ in the expression.

$- \frac{4 (3)}{9 y}$

Step 1.7.2

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

$- \frac{4 (3)}{9 (2)}$

Step 1.7.3

Cancel the common factor of $4$ and $2$ .

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Step 1.7.3.1

Factor $2$ out of $4 (3)$ .

$- \frac{2 (2 \cdot 3)}{9 (2)}$

Step 1.7.3.2

Cancel the common factors.

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Step 1.7.3.2.1

Factor $2$ out of $9 (2)$ .

$- \frac{2 (2 \cdot 3)}{2 \cdot 9}$

Step 1.7.3.2.2

Cancel the common factor.

$- \frac{2 (2 \cdot 3)}{2 \cdot 9}$

Step 1.7.3.2.3

Rewrite the expression.

$- \frac{2 \cdot 3}{9}$

Step 1.7.4

Cancel the common factor of $3$ and $9$ .

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Step 1.7.4.1

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

$- \frac{3 \cdot 2}{9}$

Step 1.7.4.2

Cancel the common factors.

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Step 1.7.4.2.1

Factor $3$ out of $9$ .

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

Step 1.7.4.2.2

Cancel the common factor.

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

Step 1.7.4.2.3

Rewrite the expression.

$- \frac{2}{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Cancel the common factor of $3$ .

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Step 2.3.1.2.3.1

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

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

Step 2.3.1.2.3.2

Factor $3$ out of $- 3$ .

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

Step 2.3.1.2.3.3

Cancel the common factor.

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

Step 2.3.1.2.3.4

Rewrite the expression.

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

Step 2.3.1.2.4

Multiply $- 2$ by $- 1$ .

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

Step 2.3.1.3

Move $2$ to the left of $x$ .

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

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.

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

Step 2.3.2.2

Add $2$ and $2$ .

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

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

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

Step 2.3.3.2

Remove parentheses.

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

Step 3