Finite Math Examples

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

Step 1

Subtract $\frac{d x}{\sqrt{x}}$ from both sides of the equation.

$\frac{d x}{\sqrt{y}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2

Simplify each term.

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Step 2.1

Multiply $\frac{d x}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{d x}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{d x}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{d x \sqrt{y}}{\sqrt{y} \sqrt{y}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{d x \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{d x \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.4

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

$\frac{d x \sqrt{y}}{{\sqrt{y}}^{1 + 1}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.5

Add $1$ and $1$ .

$\frac{d x \sqrt{y}}{{\sqrt{y}}^{2}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 2.2.6.1

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

$\frac{d x \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d x \sqrt{y}}{y^{\frac{1}{2} \cdot 2}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6.3

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

$\frac{d x \sqrt{y}}{y^{\frac{2}{2}}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$\frac{d x \sqrt{y}}{y^{\frac{2}{2}}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6.4.2

Rewrite the expression.

$\frac{d x \sqrt{y}}{y^{1}} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.2.6.5

Simplify.

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x}{\sqrt{x}} = 0$

Step 2.3

Multiply $\frac{d x}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d x \sqrt{y}}{y} + d x - (\frac{d x}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) = 0$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{d x}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{\sqrt{x} \sqrt{x}} = 0$

Step 2.4.2

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} = 0$

Step 2.4.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} = 0$

Step 2.4.4

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

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{{\sqrt{x}}^{1 + 1}} = 0$

Step 2.4.5

Add $1$ and $1$ .

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{{\sqrt{x}}^{2}} = 0$

Step 2.4.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.4.6.1

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

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} = 0$

Step 2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x^{\frac{1}{2} \cdot 2}} = 0$

Step 2.4.6.3

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

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x^{\frac{2}{2}}} = 0$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x^{\frac{2}{2}}} = 0$

Step 2.4.6.4.2

Rewrite the expression.

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x^{1}} = 0$

Step 2.4.6.5

Simplify.

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x} = 0$

Step 2.5

Cancel the common factor of $x$ .

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Step 2.5.1

Cancel the common factor.

$\frac{d x \sqrt{y}}{y} + d x - \frac{d x \sqrt{x}}{x} = 0$

Step 2.5.2

Divide $d \sqrt{x}$ by $1$ .

$\frac{d x \sqrt{y}}{y} + d x - (d \sqrt{x}) = 0$

$\frac{d x \sqrt{y}}{y} + d x - d \sqrt{x} = 0$

Step 3

Set the denominator in $\frac{d x \sqrt{y}}{y}$ equal to $0$ to find where the expression is undefined.

$y = 0$

Step 4

Set the radicand in $\sqrt{y}$ less than $0$ to find where the expression is undefined.

$y < 0$

Step 5

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$d \leq 0$

$(- \infty, 0]$

Step 7