Calculus Examples

$4 x y \frac{d y}{d x} = y^{2} - 1$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $4 x y \frac{d y}{d x} = y^{2} - 1$ by $4 x y$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $4 x y \frac{d y}{d x} = y^{2} - 1$ by $4 x y$ .

$\frac{4 x y \frac{d y}{d x}}{4 x y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

$\frac{4 x y \frac{d y}{d x}}{4 x y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2.2.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.1.2.3.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.2.3.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{y^{2}}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

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

Tap for more steps...

Step 1.1.3.1.1.1

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

$\frac{d y}{d x} = \frac{y \cdot y}{4 x y} + \frac{- 1}{4 x y}$

Step 1.1.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.3.1.1.2.1

Factor $y$ out of $4 x y$ .

$\frac{d y}{d x} = \frac{y \cdot y}{y (4 x)} + \frac{- 1}{4 x y}$

Step 1.1.3.1.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{y \cdot y}{y (4 x)} + \frac{- 1}{4 x y}$

Step 1.1.3.1.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{y}{4 x} + \frac{- 1}{4 x y}$

Step 1.1.3.1.2

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{y}{4 x} - \frac{1}{4 x y}$

Step 1.2

Factor.

Tap for more steps...

Step 1.2.1

To write $\frac{y}{4 x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{d y}{d x} = \frac{y}{4 x} \cdot \frac{y}{y} - \frac{1}{4 x y}$

Step 1.2.2

Multiply $\frac{y}{4 x}$ by $\frac{y}{y}$ .

$\frac{d y}{d x} = \frac{y \cdot y}{4 x y} - \frac{1}{4 x y}$

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{y \cdot y - 1}{4 x y}$

Step 1.2.4

Simplify the numerator.

Tap for more steps...

Step 1.2.4.1

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

$\frac{d y}{d x} = \frac{y^{1} y - 1}{4 x y}$

Step 1.2.4.2

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

$\frac{d y}{d x} = \frac{y^{1} y^{1} - 1}{4 x y}$

Step 1.2.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d y}{d x} = \frac{y^{1 + 1} - 1}{4 x y}$

Step 1.2.4.4

Add $1$ and $1$ .

$\frac{d y}{d x} = \frac{y^{2} - 1}{4 x y}$

Step 1.2.4.5

Rewrite $1$ as $1^{2}$ .

$\frac{d y}{d x} = \frac{y^{2} - 1^{2}}{4 x y}$

Step 1.2.4.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{d y}{d x} = \frac{(y + 1) (y - 1)}{4 x y}$

Step 1.3

Regroup factors.

$\frac{d y}{d x} = \frac{(y + 1) (y - 1)}{4 y} \cdot \frac{1}{x}$

Step 1.4

Multiply both sides by $\frac{4 y}{(y + 1) (y - 1)}$ .

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{4 y}{(y + 1) (y - 1)} (\frac{(y + 1) (y - 1)}{4 y} \cdot \frac{1}{x})$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Multiply $\frac{(y + 1) (y - 1)}{4 y}$ by $\frac{1}{x}$ .

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{4 y}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{4 y x}$

Step 1.5.2

Cancel the common factor of $4 y$ .

Tap for more steps...

Step 1.5.2.1

Factor $4 y$ out of $4 y x$ .

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{4 y}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{4 y (x)}$

Step 1.5.2.2

Cancel the common factor.

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{4 y}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{4 y x}$

Step 1.5.2.3

Rewrite the expression.

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{x}$

Step 1.5.3

Cancel the common factor of $(y + 1) (y - 1)$ .

Tap for more steps...

Step 1.5.3.1

Cancel the common factor.

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{x}$

Step 1.5.3.2

Rewrite the expression.

$\frac{4 y}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{x}$

Step 1.6

Rewrite the equation.

$\frac{4 y}{(y + 1) (y - 1)} d y = \frac{1}{x} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{4 y}{(y + 1) (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int \frac{y}{(y + 1) (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.2

Let $u = (y + 1) (y - 1)$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.2.1

Let $u = (y + 1) (y - 1)$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 2.2.2.1.1

Differentiate $(y + 1) (y - 1)$ .

$\frac{d}{d y} [(y + 1) (y - 1)]$

Step 2.2.2.1.2

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 $f (y) = y + 1$ and $g (y) = y - 1$ .

$(y + 1) \frac{d}{d y} [y - 1] + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3

Differentiate.

Tap for more steps...

Step 2.2.2.1.3.1

By the Sum Rule, the derivative of $y - 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$ .

$(y + 1) (\frac{d}{d y} [y] + \frac{d}{d y} [- 1]) + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3.2

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

$(y + 1) (1 + \frac{d}{d y} [- 1]) + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3.3

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

$(y + 1) (1 + 0) + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3.4

Simplify the expression.

Tap for more steps...

Step 2.2.2.1.3.4.1

Add $1$ and $0$ .

$(y + 1) \cdot 1 + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3.4.2

Multiply $y + 1$ by $1$ .

$y + 1 + (y - 1) \frac{d}{d y} [y + 1]$

Step 2.2.2.1.3.5

By the Sum Rule, the derivative of $y + 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [1]$ .

$y + 1 + (y - 1) (\frac{d}{d y} [y] + \frac{d}{d y} [1])$

Step 2.2.2.1.3.6

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

$y + 1 + (y - 1) (1 + \frac{d}{d y} [1])$

Step 2.2.2.1.3.7

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

$y + 1 + (y - 1) (1 + 0)$

Step 2.2.2.1.3.8

Simplify by adding terms.

Tap for more steps...

Step 2.2.2.1.3.8.1

Add $1$ and $0$ .

$y + 1 + (y - 1) \cdot 1$

Step 2.2.2.1.3.8.2

Multiply $y - 1$ by $1$ .

$y + 1 + y - 1$

Step 2.2.2.1.3.8.3

Add $y$ and $y$ .

$2 y + 1 - 1$

Step 2.2.2.1.3.8.4

Simplify by subtracting numbers.

Tap for more steps...

Step 2.2.2.1.3.8.4.1

Subtract $1$ from $1$ .

$2 y + 0$

Step 2.2.2.1.3.8.4.2

Add $2 y$ and $0$ .

$2 y$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$4 \int \frac{1}{u} \cdot \frac{1}{2} d u = \int \frac{1}{x} d x$

Step 2.2.3

Simplify.

Tap for more steps...

Step 2.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$4 \int \frac{1}{u \cdot 2} d u = \int \frac{1}{x} d x$

Step 2.2.3.2

Move $2$ to the left of $u$ .

$4 \int \frac{1}{2 u} d u = \int \frac{1}{x} d x$

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$4 (\frac{1}{2} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 2.2.5

Simplify.

Tap for more steps...

Step 2.2.5.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.5.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.2.5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.5.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.5.2.2.3

Rewrite the expression.

$\frac{2}{1} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.5.2.2.4

Divide $2$ by $1$ .

$2 \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.6

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$2 (\ln (| u |) + C_{1}) = \int \frac{1}{x} d x$

Step 2.2.7

Simplify.

$2 \ln (| u |) + C_{1} = \int \frac{1}{x} d x$

Step 2.2.8

Replace all occurrences of $u$ with $(y + 1) (y - 1)$ .

$2 \ln (| (y + 1) (y - 1) |) + C_{1} = \int \frac{1}{x} d x$

Step 2.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$2 \ln (| (y + 1) (y - 1) |) + C_{1} = \ln (| x |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$2 \ln (| (y + 1) (y - 1) |) = \ln (| x |) + C$