Calculus Examples

cari $\frac{d y}{d x}$ jika $4 x y - 3 y = 1$

Step 1

Write the problem as a mathematical expression.

$\frac{d y}{d x} \cdot 4 x y - 3 y = 1$

Step 2

Separate the variables.

Tap for more steps...

Step 2.1

Solve for $\frac{d y}{d x}$ .

Tap for more steps...

Step 2.1.1

Move $4$ to the left of $\frac{d y}{d x}$ .

$4 \frac{d y}{d x} x y - 3 y = 1$

Step 2.1.2

Add $3 y$ to both sides of the equation.

$4 \frac{d y}{d x} x y = 1 + 3 y$

Step 2.1.3

Divide each term in $4 \frac{d y}{d x} x y = 1 + 3 y$ by $4 x y$ and simplify.

Tap for more steps...

Step 2.1.3.1

Divide each term in $4 \frac{d y}{d x} x y = 1 + 3 y$ by $4 x y$ .

$\frac{4 \frac{d y}{d x} x y}{4 x y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2

Simplify the left side.

Tap for more steps...

Step 2.1.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.1.3.2.1.1

Cancel the common factor.

$\frac{4 \frac{d y}{d x} x y}{4 x y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2.1.2

Rewrite the expression.

$\frac{\frac{d y}{d x} x y}{x y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.1.3.2.2.1

Cancel the common factor.

$\frac{\frac{d y}{d x} x y}{x y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2.2.2

Rewrite the expression.

$\frac{\frac{d y}{d x} y}{y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 2.1.3.2.3.1

Cancel the common factor.

$\frac{\frac{d y}{d x} y}{y} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.2.3.2

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

$\frac{d y}{d x} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.3

Simplify the right side.

Tap for more steps...

Step 2.1.3.3.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 2.1.3.3.1.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.1.3.3.1.2

Rewrite the expression.

$\frac{d y}{d x} = \frac{1}{4 x y} + \frac{3}{4 x}$

Step 2.2

Factor.

Tap for more steps...

Step 2.2.1

To write $\frac{3}{4 x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{d y}{d x} = \frac{1}{4 x y} + \frac{3}{4 x} \cdot \frac{y}{y}$

Step 2.2.2

Multiply $\frac{3}{4 x}$ by $\frac{y}{y}$ .

$\frac{d y}{d x} = \frac{1}{4 x y} + \frac{3 y}{4 x y}$

Step 2.2.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{1 + 3 y}{4 x y}$

Step 2.3

Regroup factors.

$\frac{d y}{d x} = \frac{1 + 3 y}{4 y} \cdot \frac{1}{x}$

Step 2.4

Multiply both sides by $\frac{4 y}{1 + 3 y}$ .

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{4 y}{1 + 3 y} (\frac{1 + 3 y}{4 y} \cdot \frac{1}{x})$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Multiply $\frac{1 + 3 y}{4 y}$ by $\frac{1}{x}$ .

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{4 y}{1 + 3 y} \cdot \frac{1 + 3 y}{4 y x}$

Step 2.5.2

Cancel the common factor of $4 y$ .

Tap for more steps...

Step 2.5.2.1

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

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{4 y}{1 + 3 y} \cdot \frac{1 + 3 y}{4 y (x)}$

Step 2.5.2.2

Cancel the common factor.

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{4 y}{1 + 3 y} \cdot \frac{1 + 3 y}{4 y x}$

Step 2.5.2.3

Rewrite the expression.

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{1}{1 + 3 y} \cdot \frac{1 + 3 y}{x}$

Step 2.5.3

Cancel the common factor of $1 + 3 y$ .

Tap for more steps...

Step 2.5.3.1

Cancel the common factor.

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{1}{1 + 3 y} \cdot \frac{1 + 3 y}{x}$

Step 2.5.3.2

Rewrite the expression.

$\frac{4 y}{1 + 3 y} \frac{d y}{d x} = \frac{1}{x}$

Step 2.6

Rewrite the equation.

$\frac{4 y}{1 + 3 y} d y = \frac{1}{x} d x$

Step 3

Integrate both sides.

Tap for more steps...

Step 3.1

Set up an integral on each side.

$\int \frac{4 y}{1 + 3 y} d y = \int \frac{1}{x} d x$

Step 3.2

Integrate the left side.

Tap for more steps...

Step 3.2.1

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

$4 \int \frac{y}{1 + 3 y} d y = \int \frac{1}{x} d x$

Step 3.2.2

Reorder $1$ and $3 y$ .

$4 \int \frac{y}{3 y + 1} d y = \int \frac{1}{x} d x$

Step 3.2.3

Divide $y$ by $3 y + 1$ .

Tap for more steps...

Step 3.2.3.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 y$

$1$

$y$

$0$

Step 3.2.3.2

Divide the highest order term in the dividend $y$ by the highest order term in divisor $3 y$ .

					$\frac{1}{3}$
	$3 y$	+	$1$		$y$	+	$0$

Step 3.2.3.3

Multiply the new quotient term by the divisor.

				$\frac{1}{3}$
$3 y$	+	$1$		$y$	+	$0$
			+	$y$	+	$\frac{1}{3}$

Step 3.2.3.4

The expression needs to be subtracted from the dividend, so change all the signs in $y + \frac{1}{3}$

				$\frac{1}{3}$
$3 y$	+	$1$		$y$	+	$0$
			-	$y$	-	$\frac{1}{3}$

Step 3.2.3.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$\frac{1}{3}$
$3 y$	+	$1$		$y$	+	$0$
			-	$y$	-	$\frac{1}{3}$
					-	$\frac{1}{3}$

Step 3.2.3.6

The final answer is the quotient plus the remainder over the divisor.

$4 \int \frac{1}{3} - \frac{1}{3 (3 y + 1)} d y = \int \frac{1}{x} d x$

Step 3.2.4

Split the single integral into multiple integrals.

$4 (\int \frac{1}{3} d y + \int - \frac{1}{3 (3 y + 1)} d y) = \int \frac{1}{x} d x$

Step 3.2.5

Apply the constant rule.

$4 (\frac{1}{3} y + C_{1} + \int - \frac{1}{3 (3 y + 1)} d y) = \int \frac{1}{x} d x$

Step 3.2.6

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$4 (\frac{1}{3} y + C_{1} - \int \frac{1}{3 (3 y + 1)} d y) = \int \frac{1}{x} d x$

Step 3.2.7

Since $\frac{1}{3}$ is constant with respect to $y$ , move $\frac{1}{3}$ out of the integral.

$4 (\frac{1}{3} y + C_{1} - (\frac{1}{3} \int \frac{1}{3 y + 1} d y)) = \int \frac{1}{x} d x$

Step 3.2.8

Combine $\frac{1}{3}$ and $y$ .

$4 (\frac{y}{3} + C_{1} - \frac{1}{3} \int \frac{1}{3 y + 1} d y) = \int \frac{1}{x} d x$

Step 3.2.9

Let $u = 3 y + 1$ . Then $d u = 3 d y$ , so $\frac{1}{3} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 3.2.9.1

Let $u = 3 y + 1$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 3.2.9.1.1

Differentiate $3 y + 1$ .

$\frac{d}{d y} [3 y + 1]$

Step 3.2.9.1.2

By the Sum Rule, the derivative of $3 y + 1$ with respect to $y$ is $\frac{d}{d y} [3 y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [3 y] + \frac{d}{d y} [1]$

Step 3.2.9.1.3

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

Tap for more steps...

Step 3.2.9.1.3.1

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

$3 \frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 3.2.9.1.3.2

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

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

Step 3.2.9.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d y} [1]$

Step 3.2.9.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.2.9.1.4.1

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

$3 + 0$

Step 3.2.9.1.4.2

Add $3$ and $0$ .

$3$

Step 3.2.9.2

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

$4 (\frac{y}{3} + C_{1} - \frac{1}{3} \int \frac{1}{u} \cdot \frac{1}{3} d u) = \int \frac{1}{x} d x$

Step 3.2.10

Simplify.

Tap for more steps...

Step 3.2.10.1

Multiply $\frac{1}{u}$ by $\frac{1}{3}$ .

$4 (\frac{y}{3} + C_{1} - \frac{1}{3} \int \frac{1}{u \cdot 3} d u) = \int \frac{1}{x} d x$

Step 3.2.10.2

Move $3$ to the left of $u$ .

$4 (\frac{y}{3} + C_{1} - \frac{1}{3} \int \frac{1}{3 u} d u) = \int \frac{1}{x} d x$

Step 3.2.11

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$4 (\frac{y}{3} + C_{1} - \frac{1}{3} (\frac{1}{3} \int \frac{1}{u} d u)) = \int \frac{1}{x} d x$

Step 3.2.12

Simplify.

Tap for more steps...

Step 3.2.12.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$4 (\frac{y}{3} + C_{1} - \frac{1}{3 \cdot 3} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 3.2.12.2

Multiply $3$ by $3$ .

$4 (\frac{y}{3} + C_{1} - \frac{1}{9} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 3.2.13

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

$4 (\frac{y}{3} + C_{1} - \frac{1}{9} (\ln (| u |) + C_{2})) = \int \frac{1}{x} d x$

Step 3.2.14

Simplify.

$4 (\frac{1}{3} y - \frac{1}{9} \ln (| u |)) + C_{3} = \int \frac{1}{x} d x$

Step 3.2.15

Replace all occurrences of $u$ with $3 y + 1$ .

$4 (\frac{1}{3} y - \frac{1}{9} \ln (| 3 y + 1 |)) + C_{3} = \int \frac{1}{x} d x$

Step 3.3

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

$4 (\frac{1}{3} y - \frac{1}{9} \ln (| 3 y + 1 |)) + C_{3} = \ln (| x |) + C_{4}$

Step 3.4

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

$4 (\frac{1}{3} y - \frac{1}{9} \ln (| 3 y + 1 |)) = \ln (| x |) + C$