Calculus Examples

$9 x^{2} + 25 y^{2} = 225$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (9 x^{2} + 25 y^{2}) = \frac{d}{d x} (225)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{2} + 25 y^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [25 y^{2}]$ .

$\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [25 y^{2}]$

Step 2.2

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

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Step 2.2.1

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

$9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [25 y^{2}]$

Step 2.2.2

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

$9 (2 x) + \frac{d}{d x} [25 y^{2}]$

Step 2.2.3

Multiply $2$ by $9$ .

$18 x + \frac{d}{d x} [25 y^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [25 y^{2}]$ .

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Step 2.3.1

Since $25$ is constant with respect to $x$ , the derivative of $25 y^{2}$ with respect to $x$ is $25 \frac{d}{d x} [y^{2}]$ .

$18 x + 25 \frac{d}{d x} [y^{2}]$

Step 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 2.3.2.1

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

$18 x + 25 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.3.2.2

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

$18 x + 25 (2 u \frac{d}{d x} [y])$

Step 2.3.2.3

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

$18 x + 25 (2 y \frac{d}{d x} [y])$

Step 2.3.3

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

$18 x + 25 (2 y y')$

Step 2.3.4

Multiply $2$ by $25$ .

$18 x + 50 y y'$

Step 2.4

Reorder terms.

$50 y y' + 18 x$

Step 3

Since $225$ is constant with respect to $x$ , the derivative of $225$ with respect to $x$ is $0$ .

$0$

Step 4

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

$50 y y' + 18 x = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $18 x$ from both sides of the equation.

$50 y y' = - 18 x$

Step 5.2

Divide each term in $50 y y' = - 18 x$ by $50 y$ and simplify.

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Step 5.2.1

Divide each term in $50 y y' = - 18 x$ by $50 y$ .

$\frac{50 y y'}{50 y} = \frac{- 18 x}{50 y}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $50$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{50 y y'}{50 y} = \frac{- 18 x}{50 y}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 18 x}{50 y}$

Step 5.2.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 18 x}{50 y}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 18 x}{50 y}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $- 18$ and $50$ .

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Step 5.2.3.1.1

Factor $2$ out of $- 18 x$ .

$y' = \frac{2 (- 9 x)}{50 y}$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $2$ out of $50 y$ .

$y' = \frac{2 (- 9 x)}{2 (25 y)}$

Step 5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 9 x)}{2 (25 y)}$

Step 5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{- 9 x}{25 y}$

Step 5.2.3.2

Move the negative in front of the fraction.

$y' = - \frac{9 x}{25 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{9 x}{25 y}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$9 x = 0$

Step 7.2

Divide each term in $9 x = 0$ by $9$ and simplify.

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Step 7.2.1

Divide each term in $9 x = 0$ by $9$ .

$\frac{9 x}{9} = \frac{0}{9}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $9$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{0}{9}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{9}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $9$ .

$x = 0$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $9 \cdot 0^{2} + 25 y^{2}$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Raising $0$ to any positive power yields $0$ .

$9 \cdot 0 + 25 y^{2} = 225$

Step 8.1.1.2

Multiply $9$ by $0$ .

$0 + 25 y^{2} = 225$

Step 8.1.2

Add $0$ and $25 y^{2}$ .

$25 y^{2} = 225$

Step 8.2

Divide each term in $25 y^{2} = 225$ by $25$ and simplify.

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Step 8.2.1

Divide each term in $25 y^{2} = 225$ by $25$ .

$\frac{25 y^{2}}{25} = \frac{225}{25}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $25$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{25 y^{2}}{25} = \frac{225}{25}$

Step 8.2.2.1.2

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

$y^{2} = \frac{225}{25}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $225$ by $25$ .

$y^{2} = 9$

Step 8.3

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

$y = \pm \sqrt{9}$

Step 8.4

Simplify $\pm \sqrt{9}$ .

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Step 8.4.1

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

$y = \pm \sqrt{3^{2}}$

Step 8.4.2

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

$y = \pm 3$

Step 8.5

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

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Step 8.5.1

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

$y = 3$

Step 8.5.2

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

$y = - 3$

Step 8.5.3

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

$y = 3, - 3$

Step 9

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 3)$

$(0, - 3)$

Step 10