Calculus Examples

$x^{2} + x y - y^{2} = 4$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + x y - y^{2}) = \frac{d}{d x} (4)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $2$ by $- 1$ .

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

Step 2.4

Reorder terms.

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

Step 3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Subtract $y$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x - 2 y$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2

Factor $- 1$ out of $- 2 x$ .

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

Step 5.3.3.3

Factor $- 1$ out of $- y$ .

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

Step 5.3.3.4

Factor $- 1$ out of $- (2 x) - (y)$ .

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

Step 5.3.3.5

Simplify the expression.

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Step 5.3.3.5.1

Rewrite $- (2 x + y)$ as $- 1 (2 x + y)$ .

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

Step 5.3.3.5.2

Move the negative in front of the fraction.

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

Step 6

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

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