Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $x - 2$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

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

Step 2.8.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Move $x$ .

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

Step 3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 3.4.1.2

Multiply $2$ by $- 2$ .

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

Step 3.4.1.3

Multiply $- 1$ by $- 3$ .

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

Step 3.4.2

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

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

Step 3.5

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 3.5.2

Write the factored form using these integers.

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