Calculus Examples

$x^{2} + 4 y^{2} = 36$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

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

$4 y^{2} = 36 - x^{2}$

Step 1.2

Divide each term in $4 y^{2} = 36 - x^{2}$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 y^{2} = 36 - x^{2}$ by $4$ .

$\frac{4 y^{2}}{4} = \frac{36}{4} + \frac{- x^{2}}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y^{2}}{4} = \frac{36}{4} + \frac{- x^{2}}{4}$

Step 1.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{36}{4} + \frac{- x^{2}}{4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $36$ by $4$ .

$y^{2} = 9 + \frac{- x^{2}}{4}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y^{2} = 9 - \frac{x^{2}}{4}$

Step 1.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{9 - \frac{x^{2}}{4}}$

Step 1.4

Simplify $\pm \sqrt{9 - \frac{x^{2}}{4}}$ .

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Step 1.4.1

Write the expression using exponents.

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Step 1.4.1.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.4.1.2

Rewrite $\frac{x^{2}}{4}$ as ${(\frac{x}{2})}^{2}$ .

$y = \pm \sqrt{3^{2} - {(\frac{x}{2})}^{2}}$

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = \frac{x}{2}$ .

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

Step 1.4.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4.4

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

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

Step 1.4.5

Combine the numerators over the common denominator.

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

Step 1.4.6

Multiply $3$ by $2$ .

$y = \pm \sqrt{\frac{6 + x}{2} (3 - \frac{x}{2})}$

Step 1.4.7

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt{\frac{6 + x}{2} (3 \cdot \frac{2}{2} - \frac{x}{2})}$

Step 1.4.8

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

$y = \pm \sqrt{\frac{6 + x}{2} (\frac{3 \cdot 2}{2} - \frac{x}{2})}$

Step 1.4.9

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{6 + x}{2} \cdot \frac{3 \cdot 2 - x}{2}}$

Step 1.4.10

Multiply $3$ by $2$ .

$y = \pm \sqrt{\frac{6 + x}{2} \cdot \frac{6 - x}{2}}$

Step 1.4.11

Multiply $\frac{6 + x}{2}$ by $\frac{6 - x}{2}$ .

$y = \pm \sqrt{\frac{(6 + x) (6 - x)}{2 \cdot 2}}$

Step 1.4.12

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{(6 + x) (6 - x)}{4}}$

Step 1.4.13

Rewrite $\frac{(6 + x) (6 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((6 + x) (6 - x))$ .

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Step 1.4.13.1

Factor the perfect power $1^{2}$ out of $(6 + x) (6 - x)$ .

$y = \pm \sqrt{\frac{1^{2} ((6 + x) (6 - x))}{4}}$

Step 1.4.13.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{1^{2} ((6 + x) (6 - x))}{2^{2} \cdot 1}}$

Step 1.4.13.3

Rearrange the fraction $\frac{1^{2} ((6 + x) (6 - x))}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} ((6 + x) (6 - x))}$

Step 1.4.14

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{(6 + x) (6 - x)}$

Step 1.4.15

Combine $\frac{1}{2}$ and $\sqrt{(6 + x) (6 - x)}$ .

$y = \pm \frac{\sqrt{(6 + x) (6 - x)}}{2}$

Step 1.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{(6 + x) (6 - x)}}{2}$

Step 1.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$

Step 1.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = \frac{\sqrt{(6 + x) (6 - x)}}{2} \to f (x) = \frac{\sqrt{(6 + x) (6 - x)}}{2}$

$y = - \frac{\sqrt{(6 + x) (6 - x)}}{2} \to f (x) = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$

Step 3

Because the $y$ variable in the equation $x^{2} + 4 y^{2} = 36$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + 4 y^{2}) = \frac{d}{d x} (36)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

Differentiate.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.2.2.2.3

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

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

Step 3.2.2.3

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

$2 x + 4 (2 y y')$

Step 3.2.2.4

Multiply $2$ by $4$ .

$2 x + 8 y y'$

Step 3.2.3

Reorder terms.

$8 y y' + 2 x$

Step 3.3

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

$8 y y' + 2 x = 0$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Subtract $2 x$ from both sides of the equation.

$8 y y' = - 2 x$

Step 3.5.2

Divide each term in $8 y y' = - 2 x$ by $8 y$ and simplify.

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Step 3.5.2.1

Divide each term in $8 y y' = - 2 x$ by $8 y$ .

$\frac{8 y y'}{8 y} = \frac{- 2 x}{8 y}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $8$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{8 y y'}{8 y} = \frac{- 2 x}{8 y}$

Step 3.5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 2 x}{8 y}$

Step 3.5.2.2.2

Cancel the common factor of $y$ .

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Step 3.5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 2 x}{8 y}$

Step 3.5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{8 y}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Cancel the common factor of $- 2$ and $8$ .

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Step 3.5.2.3.1.1

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

$y' = \frac{2 (- x)}{8 y}$

Step 3.5.2.3.1.2

Cancel the common factors.

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Step 3.5.2.3.1.2.1

Factor $2$ out of $8 y$ .

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

Step 3.5.2.3.1.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.2.3

Rewrite the expression.

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

Step 3.5.2.3.2

Move the negative in front of the fraction.

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

Step 3.6

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

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

Step 4

Set the numerator equal to zero.

$x = 0$

Step 5

Solve the function $f (x) = \frac{\sqrt{(6 + x) (6 - x)}}{2}$ at $x = 0$ .

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{\sqrt{(6 + 0) (6 - (0))}}{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (0) = \frac{\sqrt{(6 + 0) (6 - (0))}}{2}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Add $6$ and $0$ .

$f (0) = \frac{\sqrt{6 (6 - (0))}}{2}$

Step 5.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = \frac{\sqrt{6 (6 + 0)}}{2}$

Step 5.2.2.3

Add $6$ and $0$ .

$f (0) = \frac{\sqrt{6 \cdot 6}}{2}$

Step 5.2.2.4

Multiply $6$ by $6$ .

$f (0) = \frac{\sqrt{36}}{2}$

Step 5.2.2.5

Rewrite $36$ as $6^{2}$ .

$f (0) = \frac{\sqrt{6^{2}}}{2}$

Step 5.2.2.6

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = \frac{6}{2}$

Step 5.2.3

Divide $6$ by $2$ .

$f (0) = 3$

Step 5.2.4

The final answer is $3$ .

$3$

Step 6

Solve the function $f (x) = - \frac{\sqrt{(6 + x) (6 - x)}}{2}$ at $x = 0$ .

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{\sqrt{(6 + 0) (6 - (0))}}{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f (0) = - \frac{\sqrt{(6 + 0) (6 - (0))}}{2}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Add $6$ and $0$ .

$f (0) = - \frac{\sqrt{6 (6 - (0))}}{2}$

Step 6.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = - \frac{\sqrt{6 (6 + 0)}}{2}$

Step 6.2.2.3

Add $6$ and $0$ .

$f (0) = - \frac{\sqrt{6 \cdot 6}}{2}$

Step 6.2.2.4

Multiply $6$ by $6$ .

$f (0) = - \frac{\sqrt{36}}{2}$

Step 6.2.2.5

Rewrite $36$ as $6^{2}$ .

$f (0) = - \frac{\sqrt{6^{2}}}{2}$

Step 6.2.2.6

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = - \frac{6}{2}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $6$ by $2$ .

$f (0) = - 1 \cdot 3$

Step 6.2.3.2

Multiply $- 1$ by $3$ .

$f (0) = - 3$

Step 6.2.4

The final answer is $- 3$ .

$- 3$

Step 7

The horizontal tangent lines are $y = 3, y = - 3$

$y = 3, y = - 3$

Step 8