Calculus Examples

$\frac{1 - x}{\sqrt{x} - 1}$

Step 1

Multiply $\frac{1 - x}{\sqrt{x} - 1}$ by $\frac{\sqrt{x} + 1}{\sqrt{x} + 1}$ .

$\frac{1 - x}{\sqrt{x} - 1} \cdot \frac{\sqrt{x} + 1}{\sqrt{x} + 1}$

Step 2

Multiply $\frac{1 - x}{\sqrt{x} - 1}$ by $\frac{\sqrt{x} + 1}{\sqrt{x} + 1}$ .

$\frac{(1 - x) (\sqrt{x} + 1)}{(\sqrt{x} - 1) (\sqrt{x} + 1)}$

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

$\frac{(1 - x) (\sqrt{x} + 1)}{x^{1} - 1}$

Step 4.5

Simplify.

$\frac{(1 - x) (\sqrt{x} + 1)}{x - 1}$

Step 5

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 5.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{(- 1 (- 1) - x) (\sqrt{x} + 1)}{x - 1}$

Step 5.2

Factor $- 1$ out of $- x$ .

$\frac{(- 1 (- 1) - (x)) (\sqrt{x} + 1)}{x - 1}$

Step 5.3

Factor $- 1$ out of $- 1 (- 1) - (x)$ .

$\frac{- 1 (- 1 + x) (\sqrt{x} + 1)}{x - 1}$

Step 5.4

Reorder terms.

$\frac{- 1 (x - 1) (\sqrt{x} + 1)}{x - 1}$

Step 5.5

Cancel the common factor.

$\frac{- 1 (x - 1) (\sqrt{x} + 1)}{x - 1}$

Step 5.6

Divide $- 1 (\sqrt{x} + 1)$ by $1$ .

$- 1 (\sqrt{x} + 1)$

Step 6

Rewrite $- 1 (\sqrt{x} + 1)$ as $- (\sqrt{x} + 1)$ .

$- (\sqrt{x} + 1)$

Step 7

Apply the distributive property.

$- \sqrt{x} - 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$- \sqrt{x} - 1$