Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (x) = x^{2}$ and $g (x) = y^{2}$ .

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

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 $g (x) = y$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Move $2$ to the left of $x^{2}$ .

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

Step 3.4

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

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

Step 3.5

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

$2 x^{2} y y' + y^{2} (2 x)$

Step 3.6

Move $2$ to the left of $y^{2}$ .

$2 x^{2} y y' + 2 y^{2} x$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Subtract $2 x^{2} y y'$ from both sides of the equation.

$x y' + y + 1 + y' - 2 x^{2} y y' = 2 y^{2} x$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $y$ from both sides of the equation.

$x y' + 1 + y' - 2 x^{2} y y' = 2 y^{2} x - y$

Step 5.2.2

Subtract $1$ from both sides of the equation.

$x y' + y' - 2 x^{2} y y' = 2 y^{2} x - y - 1$

Step 5.3

Factor $y'$ out of $x y' + y' - 2 x^{2} y y'$ .

Tap for more steps...

Step 5.3.1

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

$y' x + y' - 2 x^{2} y y' = 2 y^{2} x - y - 1$

Step 5.3.2

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

$y' x + y' - 2 x^{2} y y' = 2 y^{2} x - y - 1$

Step 5.3.3

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

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

Step 5.3.4

Factor $y'$ out of $- 2 x^{2} y y'$ .

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

Step 5.3.5

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

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

Step 5.3.6

Factor $y'$ out of $y' (x + 1) + y' (- 2 x^{2} y)$ .

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

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $x + 1 - 2 x^{2} y$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Simplify each term.

Tap for more steps...

Step 5.4.3.1.1

Move the negative in front of the fraction.

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

Step 5.4.3.1.2

Move the negative in front of the fraction.

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

Step 5.4.3.2

Combine into one fraction.

Tap for more steps...

Step 5.4.3.2.1

Combine the numerators over the common denominator.

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

Step 5.4.3.2.2

Combine the numerators over the common denominator.

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

Step 6

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

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