Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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 $g (y) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $x$ by $1$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $2$ by $- 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Remove unnecessary parentheses.

$x' + 2 x + 2 y x' - 2 y$

Step 2.5.3

Reorder terms.

$2 y x' + 2 x - 2 y + x'$

Step 3

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$0$

Step 4

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

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

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 $2 x$ from both sides of the equation.

$2 y x' - 2 y + x' = - 2 x$

Step 5.1.2

Add $2 y$ to both sides of the equation.

$2 y x' + x' = - 2 x + 2 y$

Step 5.2

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

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Step 5.2.1

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

$x' (2 y) + x' = - 2 x + 2 y$

Step 5.2.2

Raise $x'$ to the power of $1$ .

$x' (2 y) + x' = - 2 x + 2 y$

Step 5.2.3

Factor $x'$ out of ${x'}^{1}$ .

$x' (2 y) + x' \cdot 1 = - 2 x + 2 y$

Step 5.2.4

Factor $x'$ out of $x' (2 y) + x' \cdot 1$ .

$x' (2 y + 1) = - 2 x + 2 y$

Step 5.3

Divide each term in $x' (2 y + 1) = - 2 x + 2 y$ by $2 y + 1$ and simplify.

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Step 5.3.1

Divide each term in $x' (2 y + 1) = - 2 x + 2 y$ by $2 y + 1$ .

$\frac{x' (2 y + 1)}{2 y + 1} = \frac{- 2 x}{2 y + 1} + \frac{2 y}{2 y + 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2 y + 1$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x' (2 y + 1)}{2 y + 1} = \frac{- 2 x}{2 y + 1} + \frac{2 y}{2 y + 1}$

Step 5.3.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- 2 x}{2 y + 1} + \frac{2 y}{2 y + 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$x' = \frac{- 2 x + 2 y}{2 y + 1}$

Step 5.3.3.2

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

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Step 5.3.3.2.1

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

$x' = \frac{2 (- x) + 2 y}{2 y + 1}$

Step 5.3.3.2.2

Factor $2$ out of $2 y$ .

$x' = \frac{2 (- x) + 2 (y)}{2 y + 1}$

Step 5.3.3.2.3

Factor $2$ out of $2 (- x) + 2 (y)$ .

$x' = \frac{2 (- x + y)}{2 y + 1}$

Step 5.3.3.3

Factor $- 1$ out of $- x$ .

$x' = \frac{2 (- (x) + y)}{2 y + 1}$

Step 5.3.3.4

Factor $- 1$ out of $y$ .

$x' = \frac{2 (- (x) - 1 (- y))}{2 y + 1}$

Step 5.3.3.5

Factor $- 1$ out of $- (x) - 1 (- y)$ .

$x' = \frac{2 (- (x - y))}{2 y + 1}$

Step 5.3.3.6

Simplify the expression.

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Step 5.3.3.6.1

Rewrite $- (x - y)$ as $- 1 (x - y)$ .

$x' = \frac{2 (- 1 (x - y))}{2 y + 1}$

Step 5.3.3.6.2

Move the negative in front of the fraction.

$x' = - \frac{2 (x - y)}{2 y + 1}$

Step 6

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

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