Calculus Examples

$3 x^{2} + 2 x y + y^{2} = 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (3 x^{2} + 2 x y + y^{2}) = \frac{d}{d y} (2)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $3 x^{2} + 2 x y + y^{2}$ with respect to $y$ is $\frac{d}{d y} [3 x^{2}] + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$ .

$\frac{d}{d y} [3 x^{2}] + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2

Evaluate $\frac{d}{d y} [3 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $3$ is constant with respect to $y$ , the derivative of $3 x^{2}$ with respect to $y$ is $3 \frac{d}{d y} [x^{2}]$ .

$3 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

Tap for more steps...

Step 2.2.2.1

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

$3 (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2.2.2

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

$3 (2 u \frac{d}{d y} [x]) + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2.2.3

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

$3 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2.3

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

$3 (2 x x') + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.2.4

Multiply $2$ by $3$ .

$6 x x' + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{2}]$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$6 x x' + 2 \frac{d}{d y} [x y] + \frac{d}{d y} [y^{2}]$

Step 2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

$6 x x' + 2 (x \frac{d}{d y} [y] + y \frac{d}{d y} [x]) + \frac{d}{d y} [y^{2}]$

Step 2.3.3

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

$6 x x' + 2 (x \cdot 1 + y \frac{d}{d y} [x]) + \frac{d}{d y} [y^{2}]$

Step 2.3.4

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

$6 x x' + 2 (x \cdot 1 + y x') + \frac{d}{d y} [y^{2}]$

Step 2.3.5

Multiply $x$ by $1$ .

$6 x x' + 2 (x + y x') + \frac{d}{d y} [y^{2}]$

Step 2.4

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

$6 x x' + 2 (x + y x') + 2 y$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$6 x x' + 2 x + 2 (y x') + 2 y$

Step 2.5.2

Remove unnecessary parentheses.

$6 x x' + 2 x + 2 y x' + 2 y$

Step 2.5.3

Reorder terms.

$6 x x' + 2 y x' + 2 x + 2 y$

Step 3

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

$0$

Step 4

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

$6 x x' + 2 y x' + 2 x + 2 y = 0$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

$6 x x' + 2 y x' + 2 y = - 2 x$

Step 5.1.2

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

$6 x x' + 2 y x' = - 2 x - 2 y$

Step 5.2

Factor $2 x'$ out of $6 x x' + 2 y x'$ .

Tap for more steps...

Step 5.2.1

Factor $2 x'$ out of $6 x x'$ .

$2 x' (3 x) + 2 y x' = - 2 x - 2 y$

Step 5.2.2

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

$2 x' (3 x) + 2 x' y = - 2 x - 2 y$

Step 5.2.3

Factor $2 x'$ out of $2 x' (3 x) + 2 x' y$ .

$2 x' (3 x + y) = - 2 x - 2 y$

Step 5.3

Divide each term in $2 x' (3 x + y) = - 2 x - 2 y$ by $2 (3 x + y)$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $2 x' (3 x + y) = - 2 x - 2 y$ by $2 (3 x + y)$ .

$\frac{2 x' (3 x + y)}{2 (3 x + y)} = \frac{- 2 x}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 x' (3 x + y)}{2 (3 x + y)} = \frac{- 2 x}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x' (3 x + y)}{3 x + y} = \frac{- 2 x}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.2.2

Cancel the common factor of $3 x + y$ .

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor.

$\frac{x' (3 x + y)}{3 x + y} = \frac{- 2 x}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- 2 x}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Simplify each term.

Tap for more steps...

Step 5.3.3.1.1

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

Tap for more steps...

Step 5.3.3.1.1.1

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

$x' = \frac{2 (- x)}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.1.2.1

Cancel the common factor.

$x' = \frac{2 (- x)}{2 (3 x + y)} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$x' = \frac{- x}{3 x + y} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{x}{3 x + y} + \frac{- 2 y}{2 (3 x + y)}$

Step 5.3.3.1.3

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

Tap for more steps...

Step 5.3.3.1.3.1

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

$x' = - \frac{x}{3 x + y} + \frac{2 (- y)}{2 (3 x + y)}$

Step 5.3.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.3.2.1

Cancel the common factor.

$x' = - \frac{x}{3 x + y} + \frac{2 (- y)}{2 (3 x + y)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

$x' = - \frac{x}{3 x + y} + \frac{- y}{3 x + y}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$x' = - \frac{x}{3 x + y} - \frac{y}{3 x + y}$

Step 5.3.3.2

Simplify terms.

Tap for more steps...

Step 5.3.3.2.1

Combine the numerators over the common denominator.

$x' = \frac{- x - y}{3 x + y}$

Step 5.3.3.2.2

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

$x' = \frac{- (x) - y}{3 x + y}$

Step 5.3.3.2.3

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

$x' = \frac{- (x) - (y)}{3 x + y}$

Step 5.3.3.2.4

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

$x' = \frac{- (x + y)}{3 x + y}$

Step 5.3.3.2.5

Simplify the expression.

Tap for more steps...

Step 5.3.3.2.5.1

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

$x' = \frac{- 1 (x + y)}{3 x + y}$

Step 5.3.3.2.5.2

Move the negative in front of the fraction.

$x' = - \frac{x + y}{3 x + y}$

Step 6

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

$\frac{d x}{d y} = - \frac{x + y}{3 x + y}$