Calculus Examples

$x y + y^{2} = 1$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

$x \cdot 1 + y x' + \frac{d}{d y} [y^{2}]$

Step 2.2.4

Multiply $x$ by $1$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

$x + y x' + 2 y$

Step 2.3.2

Reorder terms.

$y x' + x + 2 y$

Step 3

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

$0$

Step 4

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

$y x' + x + 2 y = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Move all terms not containing $x'$ to the right side of the equation.

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Step 5.1.1

Subtract $x$ from both sides of the equation.

$y x' + 2 y = - x$

Step 5.1.2

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

$y x' = - x - 2 y$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $y$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

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

Step 5.2.3.1.2

Cancel the common factor of $y$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

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

Step 5.2.3.1.2.2

Divide $- 2$ by $1$ .

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

Step 6

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

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