Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

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 $g (x) = y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $y$ by $1$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.2

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

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$1 + y'$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

Subtract $2 y$ from both sides of the equation.

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

Step 5.3

Factor $y'$ out of $2 x y' + 2 y y' - y'$ .

Tap for more steps...

Step 5.3.1

Factor $y'$ out of $2 x y'$ .

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

Step 5.3.2

Factor $y'$ out of $2 y y'$ .

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

Step 5.3.3

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

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

Step 5.3.4

Factor $y'$ out of $y' (2 x) + y' (2 y)$ .

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

Step 5.3.5

Factor $y'$ out of $y' (2 x + 2 y) + y' \cdot - 1$ .

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

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

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

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Combine the numerators over the common denominator.

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

Step 6

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

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