Calculus Examples

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

Step 1

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

$2 \frac{d}{d x} [\frac{x^{2}}{2 - x^{2}}]$

Step 2

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 $f (x) = x^{2}$ and $g (x) = 2 - x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Multiply.

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 3.7.2

Multiply $x^{2}$ by $1$ .

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

Step 3.8

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

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

Step 4

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

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Step 4.1

Move $x$ .

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

Step 4.2

Multiply $x$ by $x^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $2$ .

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

Step 5

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

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

Step 6

Combine $2$ and $\frac{2 (2 - x^{2}) x + 2 x^{3}}{{(2 - x^{2})}^{2}}$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Simplify the numerator.

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Step 7.4.1

Simplify each term.

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Step 7.4.1.1

Multiply $2$ by $2$ .

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

Step 7.4.1.2

Multiply $4$ by $2$ .

$\frac{8 x + 2 (2 (- x^{2}) x) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.3

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

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Step 7.4.1.3.1

Move $x$ .

$\frac{8 x + 2 (2 (- (x \cdot x^{2}))) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 7.4.1.3.2.1

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

$\frac{8 x + 2 (2 (- (x^{1} x^{2}))) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.3.2.2

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

$\frac{8 x + 2 (2 (- x^{1 + 2})) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.3.3

Add $1$ and $2$ .

$\frac{8 x + 2 (2 (- x^{3})) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.4

Multiply $- 1$ by $2$ .

$\frac{8 x + 2 (- 2 x^{3}) + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.5

Multiply $- 2$ by $2$ .

$\frac{8 x - 4 x^{3} + 2 (2 x^{3})}{{(2 - x^{2})}^{2}}$

Step 7.4.1.6

Multiply $2$ by $2$ .

$\frac{8 x - 4 x^{3} + 4 x^{3}}{{(2 - x^{2})}^{2}}$

Step 7.4.2

Combine the opposite terms in $8 x - 4 x^{3} + 4 x^{3}$ .

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Step 7.4.2.1

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

$\frac{8 x + 0}{{(2 - x^{2})}^{2}}$

Step 7.4.2.2

Add $8 x$ and $0$ .

$\frac{8 x}{{(2 - x^{2})}^{2}}$

Step 7.5

Reorder terms.

$\frac{8 x}{{(- x^{2} + 2)}^{2}}$