Calculus Examples

$\frac{x^{3}}{1 - x^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = 1 - x^{2}$ .

$\frac{(1 - x^{2}) \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{(1 - x^{2}) (3 x^{2}) - x^{3} \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2.2

Move $3$ to the left of $1 - x^{2}$ .

$\frac{3 \cdot (1 - x^{2}) x^{2} - x^{3} \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $1 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

$\frac{3 (1 - x^{2}) x^{2} - x^{3} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{2}}$

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{3 (1 - x^{2}) x^{2} - x^{3} (0 + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{2}}$

Step 2.5

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$\frac{3 (1 - x^{2}) x^{2} - x^{3} \frac{d}{d x} [- x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{3 (1 - x^{2}) x^{2} - x^{3} (- \frac{d}{d x} [x^{2}])}{{(1 - x^{2})}^{2}}$

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

$\frac{3 (1 - x^{2}) x^{2} + 1 x^{3} \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2.7.2

Multiply $x^{3}$ by $1$ .

$\frac{3 (1 - x^{2}) x^{2} + x^{3} \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{2}}$

Step 2.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{3 (1 - x^{2}) x^{2} + x^{3} (2 x)}{{(1 - x^{2})}^{2}}$

Step 3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.1

Move $x$ .

$\frac{3 (1 - x^{2}) x^{2} + x \cdot x^{3} \cdot 2}{{(1 - x^{2})}^{2}}$

Step 3.2

Multiply $x$ by $x^{3}$ .

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Step 3.2.1

Raise $x$ to the power of $1$ .

$\frac{3 (1 - x^{2}) x^{2} + x^{1} x^{3} \cdot 2}{{(1 - x^{2})}^{2}}$

Step 3.2.2

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

$\frac{3 (1 - x^{2}) x^{2} + x^{1 + 3} \cdot 2}{{(1 - x^{2})}^{2}}$

Step 3.3

Add $1$ and $3$ .

$\frac{3 (1 - x^{2}) x^{2} + x^{4} \cdot 2}{{(1 - x^{2})}^{2}}$

Step 4

Move $2$ to the left of $x^{4}$ .

$\frac{3 (1 - x^{2}) x^{2} + 2 \cdot x^{4}}{{(1 - x^{2})}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{(3 \cdot 1 + 3 (- x^{2})) x^{2} + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{3 \cdot 1 x^{2} + 3 (- x^{2}) x^{2} + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $3$ by $1$ .

$\frac{3 x^{2} + 3 (- x^{2}) x^{2} + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.1.2.1

Move $x^{2}$ .

$\frac{3 x^{2} + 3 (- (x^{2} x^{2})) + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3.1.2.2

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

$\frac{3 x^{2} + 3 (- x^{2 + 2}) + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3.1.2.3

Add $2$ and $2$ .

$\frac{3 x^{2} + 3 (- x^{4}) + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3.1.3

Multiply $- 1$ by $3$ .

$\frac{3 x^{2} - 3 x^{4} + 2 x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.3.2

Add $- 3 x^{4}$ and $2 x^{4}$ .

$\frac{3 x^{2} - x^{4}}{{(1 - x^{2})}^{2}}$

Step 5.4

Reorder terms.

$\frac{- x^{4} + 3 x^{2}}{{(- x^{2} + 1)}^{2}}$

Step 5.5

Factor $x^{2}$ out of $- x^{4} + 3 x^{2}$ .

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Step 5.5.1

Factor $x^{2}$ out of $- x^{4}$ .

$\frac{x^{2} (- x^{2}) + 3 x^{2}}{{(- x^{2} + 1)}^{2}}$

Step 5.5.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} (- x^{2}) + x^{2} \cdot 3}{{(- x^{2} + 1)}^{2}}$

Step 5.5.3

Factor $x^{2}$ out of $x^{2} (- x^{2}) + x^{2} \cdot 3$ .

$\frac{x^{2} (- x^{2} + 3)}{{(- x^{2} + 1)}^{2}}$

Step 5.6

Simplify the denominator.

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Step 5.6.1

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

$\frac{x^{2} (- x^{2} + 3)}{{(- x^{2} + 1^{2})}^{2}}$

Step 5.6.2

Reorder $- x^{2}$ and $1^{2}$ .

$\frac{x^{2} (- x^{2} + 3)}{{(1^{2} - x^{2})}^{2}}$

Step 5.6.3

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 $b = x$ .

$\frac{x^{2} (- x^{2} + 3)}{{((1 + x) (1 - x))}^{2}}$

Step 5.6.4

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

$\frac{x^{2} (- x^{2} + 3)}{{(1 + x)}^{2} {(1 - x)}^{2}}$

Step 5.7

Factor $- 1$ out of $- x^{2}$ .

$\frac{x^{2} (- (x^{2}) + 3)}{{(1 + x)}^{2} {(1 - x)}^{2}}$

Step 5.8

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{x^{2} (- (x^{2}) - 1 (- 3))}{{(1 + x)}^{2} {(1 - x)}^{2}}$

Step 5.9

Factor $- 1$ out of $- (x^{2}) - 1 (- 3)$ .

$\frac{x^{2} (- (x^{2} - 3))}{{(1 + x)}^{2} {(1 - x)}^{2}}$

Step 5.10

Rewrite $- (x^{2} - 3)$ as $- 1 (x^{2} - 3)$ .

$\frac{x^{2} (- 1 (x^{2} - 3))}{{(1 + x)}^{2} {(1 - x)}^{2}}$

Step 5.11

Move the negative in front of the fraction.

$- \frac{x^{2} (x^{2} - 3)}{{(1 + x)}^{2} {(1 - x)}^{2}}$