Calculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = x$ .

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

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{1}{(1 + x) (1 - x)}$ .

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

Step 2.2

Apply the product rule to $(1 + x) (1 - x)$ .

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

Step 3

One to any power is one.

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