Calculus Examples

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

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{x^{2} - 3 y^{2}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{x^{2} - 3 y^{2}}]$ .

$2 \frac{d}{d x} [\frac{x}{x^{2} - 3 y^{2}}]$

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 $f (x) = x$ and $g (x) = x^{2} - 3 y^{2}$ .

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

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 $n = 1$ .

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

Step 3.2

Multiply $x^{2} - 3 y^{2}$ by $1$ .

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

Step 3.3

By the Sum Rule, the derivative of $x^{2} - 3 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 y^{2}]$ .

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

Step 3.4

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

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

Step 3.5

Since $- 3 y^{2}$ is constant with respect to $x$ , the derivative of $- 3 y^{2}$ with respect to $x$ is $0$ .

$2 \frac{x^{2} - 3 y^{2} - x (2 x + 0)}{{(x^{2} - 3 y^{2})}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2 x$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $- 1$ by $2$ .

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

Step 10.2.2

Multiply $- 3$ by $2$ .

$\frac{- 2 x^{2} - 6 y^{2}}{{(x^{2} - 3 y^{2})}^{2}}$

Step 10.3

Factor $2$ out of $- 2 x^{2} - 6 y^{2}$ .

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Step 10.3.1

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

$\frac{2 (- x^{2}) - 6 y^{2}}{{(x^{2} - 3 y^{2})}^{2}}$

Step 10.3.2

Factor $2$ out of $- 6 y^{2}$ .

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

Step 10.3.3

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

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

Step 10.4

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

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

Step 10.5

Factor $- 1$ out of $- 3 y^{2}$ .

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

Step 10.6

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

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

Step 10.7

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

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

Step 10.8

Move the negative in front of the fraction.

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