Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Regroup factors.

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

Step 1.2

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

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

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Combine.

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

Step 1.3.2

Combine.

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

Step 1.3.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.3.3.1

Cancel the common factor.

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

Step 1.3.3.2

Rewrite the expression.

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

Step 1.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.3.4.1

Cancel the common factor.

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

Step 1.3.4.2

Rewrite the expression.

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

Step 1.3.5

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

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

Step 1.3.6

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 1.3.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

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

Step 2.2.3

Simplify the answer.

Tap for more steps...

Step 2.2.3.1

Rewrite $\frac{1}{3} (\frac{1}{2} y^{2} + C_{1})$ as $\frac{1}{3} \cdot \frac{1}{2} y^{2} + C_{1}$ .

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

Step 2.2.3.2

Simplify.

Tap for more steps...

Step 2.2.3.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

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

Step 2.2.3.2.2

Multiply $3$ by $2$ .

$\frac{1}{6} y^{2} + C_{1} = \int \sec^{2} (x) d x$

Step 2.3

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$\frac{1}{6} y^{2} + C_{1} = \tan (x) + C_{2}$

Step 2.4

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

$\frac{1}{6} y^{2} = \tan (x) + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $6$ .

$6 (\frac{1}{6} y^{2}) = 6 (\tan (x) + K)$

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $6 (\frac{1}{6} y^{2})$ .

Tap for more steps...

Step 3.2.1.1.1

Combine $\frac{1}{6}$ and $y^{2}$ .

$6 \frac{y^{2}}{6} = 6 (\tan (x) + K)$

Step 3.2.1.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$6 \frac{y^{2}}{6} = 6 (\tan (x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 6 (\tan (x) + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

$y^{2} = 6 \tan (x) + 6 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{6 \tan (x) + 6 K}$

Step 3.4

Factor $6$ out of $6 \tan (x) + 6 K$ .

Tap for more steps...

Step 3.4.1

Factor $6$ out of $6 \tan (x)$ .

$y = \pm \sqrt{6 (\tan (x)) + 6 K}$

Step 3.4.2

Factor $6$ out of $6 K$ .

$y = \pm \sqrt{6 (\tan (x)) + 6 K}$

Step 3.4.3

Factor $6$ out of $6 (\tan (x)) + 6 K$ .

$y = \pm \sqrt{6 (\tan (x) + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{6 (\tan (x) + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{6 (\tan (x) + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{6 (\tan (x) + K)}$

$y = - \sqrt{6 (\tan (x) + K)}$

$y = \sqrt{6 (\tan (x) + K)}$

$y = - \sqrt{6 (\tan (x) + K)}$

$y = \sqrt{6 (\tan (x) + K)}$

$y = - \sqrt{6 (\tan (x) + K)}$