Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

$2 x + 2 y y'$

Step 2.3

Reorder terms.

$2 y y' + 2 x$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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 $g (x) = y$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $y$ by $1$ .

$2 (x y' + y)$

Step 3.6

Apply the distributive property.

$2 x y' + 2 y$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $y - x$ .

Tap for more steps...

Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Divide $y'$ by $1$ .

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

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

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.3.1.1.1

Cancel the common factor.

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

Step 5.4.3.1.1.2

Rewrite the expression.

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

Step 5.4.3.1.2

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

Tap for more steps...

Step 5.4.3.1.2.1

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

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

Step 5.4.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.2.2.1

Cancel the common factor.

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

Step 5.4.3.1.2.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Move the negative in front of the fraction.

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

Step 5.4.3.2

Simplify terms.

Tap for more steps...

Step 5.4.3.2.1

Combine the numerators over the common denominator.

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

Step 5.4.3.2.2

Cancel the common factor of $y - x$ .

Tap for more steps...

Step 5.4.3.2.2.1

Cancel the common factor.

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

Step 5.4.3.2.2.2

Rewrite the expression.

$y' = 1$

Step 6

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

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