Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $4 x y - x^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [4 x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [4 x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x y]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x y$ with respect to $x$ is $4 \frac{d}{d x} [x y]$ .

$4 \frac{d}{d x} [x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$

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 $g (x) = y$ .

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

Step 2.2.3

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

$4 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$

Step 2.2.4

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

$4 (x y' + y \cdot 1) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$

Step 2.2.5

Multiply $y$ by $1$ .

$4 (x y' + y) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 2]$

Step 2.3

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

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Step 2.3.1

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

$4 (x y' + y) - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2]$

Step 2.3.2

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

$4 (x y' + y) - (2 x) + \frac{d}{d x} [- 2]$

Step 2.3.3

Multiply $2$ by $- 1$ .

$4 (x y' + y) - 2 x + \frac{d}{d x} [- 2]$

Step 2.4

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

$4 (x y' + y) - 2 x + 0$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$4 (x y') + 4 y - 2 x + 0$

Step 2.5.2

Add $4 x y' + 4 y - 2 x$ and $0$ .

$4 x y' + 4 y - 2 x$

Step 2.5.3

Reorder terms.

$4 x y' - 2 x + 4 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 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 3.2.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Multiply $2$ by $2$ .

$4 (y \frac{d}{d x} [y])$

Step 3.4

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

$4 y y'$

Step 4

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

$4 x y' - 2 x + 4 y = 4 y y'$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $4 y y'$ from both sides of the equation.

$4 x y' - 2 x + 4 y - 4 y y' = 0$

Step 5.2

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

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Step 5.2.1

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

$4 x y' + 4 y - 4 y y' = 2 x$

Step 5.2.2

Subtract $4 y$ from both sides of the equation.

$4 x y' - 4 y y' = 2 x - 4 y$

Step 5.3

Factor $4 y'$ out of $4 x y' - 4 y y'$ .

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Step 5.3.1

Factor $4 y'$ out of $4 x y'$ .

$4 y' x - 4 y y' = 2 x - 4 y$

Step 5.3.2

Factor $4 y'$ out of $- 4 y y'$ .

$4 y' x + 4 y' (- y) = 2 x - 4 y$

Step 5.3.3

Factor $4 y'$ out of $4 y' x + 4 y' (- y)$ .

$4 y' (x - y) = 2 x - 4 y$

Step 5.4

Divide each term in $4 y' (x - y) = 2 x - 4 y$ by $4 (x - y)$ and simplify.

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Step 5.4.1

Divide each term in $4 y' (x - y) = 2 x - 4 y$ by $4 (x - y)$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $4$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $x - y$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $2$ and $4$ .

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Step 5.4.3.1.1.1

Factor $2$ out of $2 x$ .

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

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $2$ out of $4 (x - y)$ .

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

Step 5.4.3.1.1.2.2

Cancel the common factor.

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

Step 5.4.3.1.1.2.3

Rewrite the expression.

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

Step 5.4.3.1.2

Cancel the common factor of $- 4$ and $4$ .

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Step 5.4.3.1.2.1

Factor $4$ out of $- 4 y$ .

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

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Cancel the common factor.

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

Step 5.4.3.1.2.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Move the negative in front of the fraction.

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

Step 5.4.3.2

To write $- \frac{y}{x - y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.4.3.3

Write each expression with a common denominator of $2 (x - y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{y}{x - y}$ by $\frac{2}{2}$ .

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

Step 5.4.3.3.2

Reorder the factors of $(x - y) \cdot 2$ .

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

Multiply $2$ by $- 1$ .

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

Step 6

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

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