Calculus Examples

$\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$

Step 1

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

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Step 1.1

Subtract $\frac{x^{2}}{16}$ from both sides of the equation.

$\frac{y^{2}}{4} = 1 - \frac{x^{2}}{16}$

Step 1.2

Multiply both sides of the equation by $4$ .

$4 \frac{y^{2}}{4} = 4 (1 - \frac{x^{2}}{16})$

Step 1.3

Simplify both sides of the equation.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Cancel the common factor of $4$ .

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Step 1.3.1.1.1

Cancel the common factor.

$4 \frac{y^{2}}{4} = 4 (1 - \frac{x^{2}}{16})$

Step 1.3.1.1.2

Rewrite the expression.

$y^{2} = 4 (1 - \frac{x^{2}}{16})$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $4 (1 - \frac{x^{2}}{16})$ .

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Step 1.3.2.1.1

Apply the distributive property.

$y^{2} = 4 \cdot 1 + 4 (- \frac{x^{2}}{16})$

Step 1.3.2.1.2

Multiply $4$ by $1$ .

$y^{2} = 4 + 4 (- \frac{x^{2}}{16})$

Step 1.3.2.1.3

Cancel the common factor of $4$ .

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Step 1.3.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{16}$ into the numerator.

$y^{2} = 4 + 4 \frac{- x^{2}}{16}$

Step 1.3.2.1.3.2

Factor $4$ out of $16$ .

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

Step 1.3.2.1.3.3

Cancel the common factor.

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

Step 1.3.2.1.3.4

Rewrite the expression.

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

Step 1.3.2.1.4

Move the negative in front of the fraction.

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Write the expression using exponents.

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Step 1.5.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.5.1.2

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Combine the numerators over the common denominator.

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

Step 1.5.6

Multiply $2$ by $2$ .

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

Step 1.5.7

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

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

Step 1.5.8

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

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

Step 1.5.9

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 + x}{2} \cdot \frac{2 \cdot 2 - x}{2}}$

Step 1.5.10

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{4 + x}{2} \cdot \frac{4 - x}{2}}$

Step 1.5.11

Multiply $\frac{4 + x}{2}$ by $\frac{4 - x}{2}$ .

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

Step 1.5.12

Multiply $2$ by $2$ .

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

Step 1.5.13

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

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Step 1.5.13.1

Factor the perfect power $1^{2}$ out of $(4 + x) (4 - x)$ .

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

Step 1.5.13.2

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

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

Step 1.5.13.3

Rearrange the fraction $\frac{1^{2} ((4 + x) (4 - x))}{2^{2} \cdot 1}$ .

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

Step 1.5.14

Pull terms out from under the radical.

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

Step 1.5.15

Combine $\frac{1}{2}$ and $\sqrt{(4 + x) (4 - x)}$ .

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

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

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

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

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

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

Step 2

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

$y = \frac{\sqrt{(4 + x) (4 - x)}}{2} \to f (x) = \frac{\sqrt{(4 + x) (4 - x)}}{2}$

$y = - \frac{\sqrt{(4 + x) (4 - x)}}{2} \to f (x) = - \frac{\sqrt{(4 + x) (4 - x)}}{2}$

Step 3

Because the $y$ variable in the equation $\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$ 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.

$\frac{d}{d x} (\frac{x^{2}}{16} + \frac{y^{2}}{4}) = \frac{d}{d x} (1)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

By the Sum Rule, the derivative of $\frac{x^{2}}{16} + \frac{y^{2}}{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{16}] + \frac{d}{d x} [\frac{y^{2}}{4}]$ .

$\frac{d}{d x} [\frac{x^{2}}{16}] + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2

Evaluate $\frac{d}{d x} [\frac{x^{2}}{16}]$ .

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Step 3.2.2.1

Since $\frac{1}{16}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{16}$ with respect to $x$ is $\frac{1}{16} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{16} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.2

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

$\frac{1}{16} (2 x) + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.3

Combine $2$ and $\frac{1}{16}$ .

$\frac{2}{16} x + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.4

Combine $\frac{2}{16}$ and $x$ .

$\frac{2 x}{16} + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.5

Cancel the common factor of $2$ and $16$ .

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Step 3.2.2.5.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{16} + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.5.2

Cancel the common factors.

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Step 3.2.2.5.2.1

Factor $2$ out of $16$ .

$\frac{2 x}{2 \cdot 8} + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.5.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 8} + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.2.5.2.3

Rewrite the expression.

$\frac{x}{8} + \frac{d}{d x} [\frac{y^{2}}{4}]$

Step 3.2.3

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

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Step 3.2.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{y^{2}}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [y^{2}]$ .

$\frac{x}{8} + \frac{1}{4} \frac{d}{d x} [y^{2}]$

Step 3.2.3.2

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 3.2.3.2.1

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

$\frac{x}{8} + \frac{1}{4} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 3.2.3.2.2

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

$\frac{x}{8} + \frac{1}{4} (2 u \frac{d}{d x} [y])$

Step 3.2.3.2.3

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

$\frac{x}{8} + \frac{1}{4} (2 y \frac{d}{d x} [y])$

Step 3.2.3.3

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

$\frac{x}{8} + \frac{1}{4} (2 y y')$

Step 3.2.3.4

Combine $2$ and $\frac{1}{4}$ .

$\frac{x}{8} + \frac{2}{4} (y y')$

Step 3.2.3.5

Combine $y$ and $\frac{2}{4}$ .

$\frac{x}{8} + \frac{y \cdot 2}{4} y'$

Step 3.2.3.6

Combine $\frac{y \cdot 2}{4}$ and $y'$ .

$\frac{x}{8} + \frac{y \cdot 2 y'}{4}$

Step 3.2.3.7

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

$\frac{x}{8} + \frac{2 \cdot y y'}{4}$

Step 3.2.3.8

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

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Step 3.2.3.8.1

Factor $2$ out of $2 y y'$ .

$\frac{x}{8} + \frac{2 (y y')}{4}$

Step 3.2.3.8.2

Cancel the common factors.

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Step 3.2.3.8.2.1

Factor $2$ out of $4$ .

$\frac{x}{8} + \frac{2 (y y')}{2 \cdot 2}$

Step 3.2.3.8.2.2

Cancel the common factor.

$\frac{x}{8} + \frac{2 (y y')}{2 \cdot 2}$

Step 3.2.3.8.2.3

Rewrite the expression.

$\frac{x}{8} + \frac{y y'}{2}$

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

$\frac{x}{8} + \frac{y y'}{2} = 0$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Subtract $\frac{x}{8}$ from both sides of the equation.

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

Step 3.5.2

Multiply both sides of the equation by $2$ .

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

Step 3.5.3

Simplify both sides of the equation.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Cancel the common factor of $2$ .

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Step 3.5.3.1.1.1

Cancel the common factor.

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

Step 3.5.3.1.1.2

Rewrite the expression.

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

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Simplify $2 (- \frac{x}{8})$ .

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Step 3.5.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.5.3.2.1.1.1

Move the leading negative in $- \frac{x}{8}$ into the numerator.

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

Step 3.5.3.2.1.1.2

Factor $2$ out of $8$ .

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

Step 3.5.3.2.1.1.3

Cancel the common factor.

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

Step 3.5.3.2.1.1.4

Rewrite the expression.

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

Step 3.5.3.2.1.2

Move the negative in front of the fraction.

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

Step 3.5.4

Divide each term in $y y' = - \frac{x}{4}$ by $y$ and simplify.

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Step 3.5.4.1

Divide each term in $y y' = - \frac{x}{4}$ by $y$ .

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $y$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{x}{4} \cdot \frac{1}{y}$

Step 3.5.4.3.2

Multiply $\frac{1}{y}$ by $\frac{x}{4}$ .

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

Step 3.5.4.3.3

Move $4$ to the left of $y$ .

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

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 $f (x) = \frac{\sqrt{(4 + x) (4 - x)}}{2}$ at $x = 0$ .

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Step 5.1

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

$f (0) = \frac{\sqrt{(4 + 0) (4 - (0))}}{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (0) = \frac{\sqrt{(4 + 0) (4 - (0))}}{2}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Add $4$ and $0$ .

$f (0) = \frac{\sqrt{4 (4 - (0))}}{2}$

Step 5.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = \frac{\sqrt{4 (4 + 0)}}{2}$

Step 5.2.2.3

Add $4$ and $0$ .

$f (0) = \frac{\sqrt{4 \cdot 4}}{2}$

Step 5.2.2.4

Multiply $4$ by $4$ .

$f (0) = \frac{\sqrt{16}}{2}$

Step 5.2.2.5

Rewrite $16$ as $4^{2}$ .

$f (0) = \frac{\sqrt{4^{2}}}{2}$

Step 5.2.2.6

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

$f (0) = \frac{4}{2}$

Step 5.2.3

Divide $4$ by $2$ .

$f (0) = 2$

Step 5.2.4

The final answer is $2$ .

$2$

Step 6

Solve the function $f (x) = - \frac{\sqrt{(4 + x) (4 - x)}}{2}$ at $x = 0$ .

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Step 6.1

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

$f (0) = - \frac{\sqrt{(4 + 0) (4 - (0))}}{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f (0) = - \frac{\sqrt{(4 + 0) (4 - (0))}}{2}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Add $4$ and $0$ .

$f (0) = - \frac{\sqrt{4 (4 - (0))}}{2}$

Step 6.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = - \frac{\sqrt{4 (4 + 0)}}{2}$

Step 6.2.2.3

Add $4$ and $0$ .

$f (0) = - \frac{\sqrt{4 \cdot 4}}{2}$

Step 6.2.2.4

Multiply $4$ by $4$ .

$f (0) = - \frac{\sqrt{16}}{2}$

Step 6.2.2.5

Rewrite $16$ as $4^{2}$ .

$f (0) = - \frac{\sqrt{4^{2}}}{2}$

Step 6.2.2.6

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

$f (0) = - \frac{4}{2}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $4$ by $2$ .

$f (0) = - 1 \cdot 2$

Step 6.2.3.2

Multiply $- 1$ by $2$ .

$f (0) = - 2$

Step 6.2.4

The final answer is $- 2$ .

$- 2$

Step 7

The horizontal tangent lines are $y = 2, y = - 2$

$y = 2, y = - 2$

Step 8