Calculus Examples

$x^{2} + 4 x y + 4 y = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + 4 x y + 4 y) = \frac{d}{d x} (1)$

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 $x^{2} + 4 x y + 4 y$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x y] + \frac{d}{d x} [4 y]$ .

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

Step 2.1.2

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

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

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

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

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 $g (x) = y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $y$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Remove unnecessary parentheses.

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

Step 2.4.3

Reorder terms.

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

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.

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

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.

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

Step 5.1.2

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

$4 y' x + 4 y' \cdot 1 = - 2 x - 4 y$

Step 5.2.3

Factor $4 y'$ out of $4 y' x + 4 y' \cdot 1$ .

$4 y' (x + 1) = - 2 x - 4 y$

Step 5.3

Divide each term in $4 y' (x + 1) = - 2 x - 4 y$ by $4 (x + 1)$ and simplify.

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Step 5.3.1

Divide each term in $4 y' (x + 1) = - 2 x - 4 y$ by $4 (x + 1)$ .

$\frac{4 y' (x + 1)}{4 (x + 1)} = \frac{- 2 x}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $4$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{4 y' (x + 1)}{4 (x + 1)} = \frac{- 2 x}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x + 1)}{x + 1} = \frac{- 2 x}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.2.2

Cancel the common factor of $x + 1$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x + 1)}{x + 1} = \frac{- 2 x}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

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

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Step 5.3.3.1.1.1

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

$y' = \frac{2 (- x)}{4 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Factor $2$ out of $4 (x + 1)$ .

$y' = \frac{2 (- x)}{2 (2 (x + 1))} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- x)}{2 (2 (x + 1))} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- x}{2 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{x}{2 (x + 1)} + \frac{- 4 y}{4 (x + 1)}$

Step 5.3.3.1.3

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

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Step 5.3.3.1.3.1

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

$y' = - \frac{x}{2 (x + 1)} + \frac{4 (- y)}{4 (x + 1)}$

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{x}{2 (x + 1)} + \frac{4 (- y)}{4 (x + 1)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.3.1.4

Move the negative in front of the fraction.

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

Step 5.3.3.2

To write $- \frac{y}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.3.3

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

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Step 5.3.3.3.1

Multiply $\frac{y}{x + 1}$ by $\frac{2}{2}$ .

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

Step 5.3.3.3.2

Reorder the factors of $(x + 1) \cdot 2$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Multiply $2$ by $- 1$ .

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

Step 5.3.3.6

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

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

Step 5.3.3.7

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

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

Step 5.3.3.8

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

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

Step 5.3.3.9

Simplify the expression.

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Step 5.3.3.9.1

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

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

Step 5.3.3.9.2

Move the negative in front of the fraction.

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

Step 6

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

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