Calculus Examples

$x y + x = 2$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

Step 2.3

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

$x y' + y + 1$

Step 3

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

$0$

Step 4

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

$x y' + y + 1 = 0$

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 $y$ from both sides of the equation.

$x y' + 1 = - y$

Step 5.1.2

Subtract $1$ from both sides of the equation.

$x y' = - y - 1$

Step 5.2

Divide each term in $x y' = - y - 1$ by $x$ and simplify.

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Step 5.2.1

Divide each term in $x y' = - y - 1$ by $x$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $x$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

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

Step 5.2.3.1.2

Move the negative in front of the fraction.

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

Step 6

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

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