Calculus Examples

$y (x + 1) + x^{2} = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y (x + 1) + x^{2}) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $y (x + 1) + x^{2}$ with respect to $x$ is $\frac{d}{d x} [y (x + 1)] + \frac{d}{d x} [x^{2}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [y (x + 1)]$ .

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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 $f (x) = y$ and $g (x) = x + 1$ .

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

Step 2.2.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$y (1 + 0) + (x + 1) \frac{d}{d x} [y] + \frac{d}{d x} [x^{2}]$

Step 2.2.5

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

$y (1 + 0) + (x + 1) y' + \frac{d}{d x} [x^{2}]$

Step 2.2.6

Add $1$ and $0$ .

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

Step 2.2.7

Multiply $y$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$y + x y' + 1 y' + 2 x$

Step 2.4.2

Multiply $y'$ by $1$ .

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

Step 2.4.3

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

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.

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

Step 5.1.2

Subtract $y$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x + 1$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

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

Move the negative in front of the fraction.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2.2

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

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

Step 5.3.3.2.3

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

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

Step 5.3.3.2.4

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

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

Step 5.3.3.2.5

Simplify the expression.

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Step 5.3.3.2.5.1

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

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

Step 5.3.3.2.5.2

Move the negative in front of the fraction.

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

Step 6

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

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