Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Apply the product rule to $\frac{\sec (x)}{y}$ .

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

Step 1.2

Multiply both sides by $y^{2}$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

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

Step 2.3

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

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

Step 2.4

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

$\frac{1}{3} y^{3} = \tan (x) + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\tan (x) + K)$

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 $3 (\frac{1}{3} y^{3})$ .

Tap for more steps...

Step 3.2.1.1.1

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

$3 \frac{y^{3}}{3} = 3 (\tan (x) + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (\tan (x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\tan (x) + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

$y^{3} = 3 \tan (x) + 3 K$

Step 3.3

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

$y = \sqrt[3]{3 \tan (x) + 3 K}$

Step 3.4

Factor $3$ out of $3 \tan (x) + 3 K$ .

Tap for more steps...

Step 3.4.1

Factor $3$ out of $3 \tan (x)$ .

$y = \sqrt[3]{3 (\tan (x)) + 3 K}$

Step 3.4.2

Factor $3$ out of $3 (\tan (x)) + 3 K$ .

$y = \sqrt[3]{3 (\tan (x) + K)}$