Calculus Examples

$162 = y x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (162) = \frac{d}{d x} (y x^{2})$

Step 2

Since $162$ is constant with respect to $x$ , the derivative of $162$ with respect to $x$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = y$ and $g (x) = x^{2}$ .

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

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

$2 y x + x^{2} y'$

Step 3.4

Reorder terms.

$2 x y + x^{2} y'$

Step 4

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

$0 = 2 x y + x^{2} y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $2 x y + x^{2} y' = 0$ .

$2 x y + x^{2} y' = 0$

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $x$ out of $x^{2}$ .

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

Step 5.3.3.1.2.2

Cancel the common factor.

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

Step 5.3.3.1.2.3

Rewrite the expression.

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

Step 5.3.3.2

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{2 y}{x}$