Calculus Examples

$1 = 3 x + 2 x^{2} y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (1) = \frac{d}{d x} (3 x + 2 x^{2} y^{2})$

Step 2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

$3 \cdot 1 + \frac{d}{d x} [2 x^{2} y^{2}]$

Step 3.2.3

Multiply $3$ by $1$ .

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

$3 + 2 (x^{2} (2 y y') + y^{2} (2 x))$

Step 3.3.6

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

$3 + 2 (2 \cdot x^{2} y y' + y^{2} (2 x))$

Step 3.3.7

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

$3 + 2 (2 x^{2} y y' + 2 y^{2} x)$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

$3 + 2 (2 x^{2} y y') + 2 (2 y^{2} x)$

Step 3.4.2

Combine terms.

Tap for more steps...

Step 3.4.2.1

Multiply $2$ by $2$ .

$3 + 4 (x^{2} y y') + 2 (2 y^{2} x)$

Step 3.4.2.2

Multiply $2$ by $2$ .

$3 + 4 x^{2} y y' + 4 y^{2} x$

Step 3.4.3

Reorder terms.

$4 x^{2} y y' + 4 y^{2} x + 3$

Step 4

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

$0 = 4 x^{2} y y' + 4 y^{2} x + 3$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $4 x^{2} y y' + 4 y^{2} x + 3 = 0$ .

$4 x^{2} y y' + 4 y^{2} x + 3 = 0$

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 $4 y^{2} x$ from both sides of the equation.

$4 x^{2} y y' + 3 = - 4 y^{2} x$

Step 5.2.2

Subtract $3$ from both sides of the equation.

$4 x^{2} y y' = - 4 y^{2} x - 3$

Step 5.3

Divide each term in $4 x^{2} y y' = - 4 y^{2} x - 3$ by $4 x^{2} y$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $4 x^{2} y y' = - 4 y^{2} x - 3$ by $4 x^{2} y$ .

$\frac{4 x^{2} y y'}{4 x^{2} y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{4 x^{2} y y'}{4 x^{2} y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x^{2} y y'}{x^{2} y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor.

$\frac{x^{2} y y'}{x^{2} y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.3.2.3.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 4 y^{2} x}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

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 $- 4$ and $4$ .

Tap for more steps...

Step 5.3.3.1.1.1

Factor $4$ out of $- 4 y^{2} x$ .

$y' = \frac{4 (- y^{2} x)}{4 x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.1.2.1

Factor $4$ out of $4 x^{2} y$ .

$y' = \frac{4 (- y^{2} x)}{4 (x^{2} y)} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{4 (- y^{2} x)}{4 (x^{2} y)} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- y^{2} x}{x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.2

Cancel the common factor of $y^{2}$ and $y$ .

Tap for more steps...

Step 5.3.3.1.2.1

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

$y' = \frac{y (- y x)}{x^{2} y} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.2.2.1

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

$y' = \frac{y (- y x)}{y x^{2}} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.2.2.2

Cancel the common factor.

$y' = \frac{y (- y x)}{y x^{2}} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.2.2.3

Rewrite the expression.

$y' = \frac{- y x}{x^{2}} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.3

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 5.3.3.1.3.1

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

$y' = \frac{x (- y)}{x^{2}} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.3.2.1

Factor $x$ out of $x^{2}$ .

$y' = \frac{x (- y)}{x \cdot x} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.3.2.2

Cancel the common factor.

$y' = \frac{x (- y)}{x \cdot x} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.3.2.3

Rewrite the expression.

$y' = \frac{- y}{x} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{y}{x} + \frac{- 3}{4 x^{2} y}$

Step 5.3.3.1.5

Move the negative in front of the fraction.

$y' = - \frac{y}{x} - \frac{3}{4 x^{2} y}$

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x} - \frac{3}{4 x^{2} y}$