Calculus Examples

$\frac{d y}{d x} = \frac{\sqrt{x} + x}{\sqrt{y} - y}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $\sqrt{y} - y$ .

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Simplify the numerator.

Tap for more steps...

Step 1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.2.1.2

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} + x$ .

Tap for more steps...

Step 1.2.1.2.1

Multiply by $1$ .

$(\sqrt{y} - y) \frac{d y}{d x} = (\sqrt{y} - y) \frac{x^{\frac{1}{2}} \cdot 1 + x}{\sqrt{y} - y}$

Step 1.2.1.2.2

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

$(\sqrt{y} - y) \frac{d y}{d x} = (\sqrt{y} - y) \frac{x^{\frac{1}{2}} \cdot 1 + x^{1}}{\sqrt{y} - y}$

Step 1.2.1.2.3

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$(\sqrt{y} - y) \frac{d y}{d x} = (\sqrt{y} - y) \frac{x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} x^{\frac{1}{2}}}{\sqrt{y} - y}$

Step 1.2.1.2.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 1.2.2

Simplify the denominator.

Tap for more steps...

Step 1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 1.2.2.2

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} - y$ .

Tap for more steps...

Step 1.2.2.2.1

Multiply by $1$ .

$(\sqrt{y} - y) \frac{d y}{d x} = (\sqrt{y} - y) \frac{x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{y^{\frac{1}{2}} \cdot 1 - y}$

Step 1.2.2.2.2

Factor $y^{\frac{1}{2}}$ out of $- y$ .

$(\sqrt{y} - y) \frac{d y}{d x} = (\sqrt{y} - y) \frac{x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{y^{\frac{1}{2}} \cdot 1 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})}$

Step 1.2.2.2.3

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} \cdot 1 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

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

Step 1.2.3

Multiply $\sqrt{y} - y$ by $\frac{x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{y^{\frac{1}{2}} (1 - y^{\frac{1}{2}})}$ .

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

Step 1.2.4

Simplify the numerator.

Tap for more steps...

Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 1.2.4.2

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} - y$ .

Tap for more steps...

Step 1.2.4.2.1

Multiply by $1$ .

$(\sqrt{y} - y) \frac{d y}{d x} = \frac{(y^{\frac{1}{2}} \cdot 1 - y) x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{y^{\frac{1}{2}} (1 - y^{\frac{1}{2}})}$

Step 1.2.4.2.2

Factor $y^{\frac{1}{2}}$ out of $- y$ .

$(\sqrt{y} - y) \frac{d y}{d x} = \frac{(y^{\frac{1}{2}} \cdot 1 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})) x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{y^{\frac{1}{2}} (1 - y^{\frac{1}{2}})}$

Step 1.2.4.2.3

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} \cdot 1 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

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

Step 1.2.5

Cancel the common factor.

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

Step 1.2.6

Rewrite the expression.

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

Step 1.2.7

Cancel the common factor.

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

Step 1.2.8

Divide $x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})$ by $1$ .

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

Step 1.2.9

Apply the distributive property.

$(\sqrt{y} - y) \frac{d y}{d x} = x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} x^{\frac{1}{2}}$

Step 1.2.10

Multiply $x^{\frac{1}{2}}$ by $1$ .

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

Step 1.2.11

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 1.2.11.1

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

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

Step 1.2.11.2

Combine the numerators over the common denominator.

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

Step 1.2.11.3

Add $1$ and $1$ .

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

Step 1.2.11.4

Divide $2$ by $2$ .

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

Step 1.2.12

Simplify $x^{1}$ .

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

$\frac{2}{3} y^{\frac{3}{2}} - \frac{1}{2} y^{2} + C_{3} = \frac{2}{3} x^{\frac{3}{2}} + C_{4} + \int x d x$

Step 2.3.3

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

$\frac{2}{3} y^{\frac{3}{2}} - \frac{1}{2} y^{2} + C_{3} = \frac{2}{3} x^{\frac{3}{2}} + C_{4} + \frac{1}{2} x^{2} + C_{5}$

Step 2.3.4

Simplify.

$\frac{2}{3} y^{\frac{3}{2}} - \frac{1}{2} y^{2} + C_{3} = \frac{2}{3} x^{\frac{3}{2}} + \frac{1}{2} x^{2} + C_{6}$

Step 2.4

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

$\frac{2}{3} y^{\frac{3}{2}} - \frac{1}{2} y^{2} = \frac{2}{3} x^{\frac{3}{2}} + \frac{1}{2} x^{2} + K$