Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $f (x) = x$ and $g (x) = y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $y$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $2$ by $- 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $x - y$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

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

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2

Rewrite the expression.

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

Step 5.3.3.2

Move the negative in front of the fraction.

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

Step 6

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

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