Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

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 $b = x$ .

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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 $b = x$ .

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

Step 1.4

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

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Add $1$ and $1$ .

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

Step 1.5.6

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

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Step 1.5.6.1

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

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

Step 1.5.6.2

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

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

Step 1.5.6.3

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

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

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

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

Step 1.5.6.4.2

Rewrite the expression.

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

Step 1.5.6.5

Simplify.

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

Step 2

To write $\sqrt{(1 + x) (1 - x)}$ as a fraction with a common denominator, multiply by $\frac{(1 + x) (1 - x)}{(1 + x) (1 - x)}$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $\sqrt{(1 + x) (1 - x)}$ out of $\sqrt{(1 + x) (1 - x)} ((1 + x) (1 - x)) - x^{2} \sqrt{(1 + x) (1 - x)}$ .

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Step 4.1.1

Factor $\sqrt{(1 + x) (1 - x)}$ out of $- x^{2} \sqrt{(1 + x) (1 - x)}$ .

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

Step 4.1.2

Factor $\sqrt{(1 + x) (1 - x)}$ out of $\sqrt{(1 + x) (1 - x)} ((1 + x) (1 - x)) + \sqrt{(1 + x) (1 - x)} (- x^{2})$ .

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

Step 4.2

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2

Multiply $- x$ by $1$ .

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

Step 4.3.1.3

Multiply $x$ by $1$ .

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

Step 4.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.5.1

Move $x$ .

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

Step 4.3.1.5.2

Multiply $x$ by $x$ .

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

Step 4.3.2

Add $- x$ and $x$ .

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

Step 4.3.3

Add $1$ and $0$ .

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

Step 4.4

Subtract $x^{2}$ from $- x^{2}$ .

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