Calculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 2 x - 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Subtract $1$ from $1$ .

$\frac{2}{\sqrt{(2 x + 0) (1 - (2 x - 1))}}$

Step 1.3.2

Add $2 x$ and $0$ .

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Multiply $2$ by $- 1$ .

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

Step 1.3.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.6

Add $1$ and $1$ .

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

Step 1.3.7

Factor $2$ out of $- 2 x + 2$ .

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Step 1.3.7.1

Factor $2$ out of $- 2 x$ .

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

Step 1.3.7.2

Factor $2$ out of $2$ .

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

Step 1.3.7.3

Factor $2$ out of $2 (- x) + 2 (1)$ .

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

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

Step 1.3.8

Multiply $2$ by $2$ .

$\frac{2}{\sqrt{4 x (- x + 1)}}$

Step 1.4

Rewrite $4 x (- x + 1)$ as $2^{2} (x (- x + 1^{2}))$ .

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Step 1.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.4.2

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

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

Step 1.4.3

Add parentheses.

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

Step 1.5

Pull terms out from under the radical.

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

Step 1.6

One to any power is one.

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

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

Raise $\sqrt{x (- x + 1)}$ to the power of $1$ .

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

Step 4.3

Raise $\sqrt{x (- x + 1)}$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{x (- x + 1)}}^{2}$ as $x (- x + 1)$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Rewrite negatives.

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Step 8.1

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

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

Step 8.2

Move the negative in front of the fraction.

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