Calculus Examples

$\frac{x^{2} - x - 6}{x - 3}$

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 $f (x) = x^{2} - x - 6$ and $g (x) = x - 3$ .

$\frac{(x - 3) \frac{d}{d x} [x^{2} - x - 6] - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$\frac{(x - 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.2

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

$\frac{(x - 3) (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.3

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

$\frac{(x - 3) (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 6]) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.4

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

$\frac{(x - 3) (2 x - 1 \cdot 1 + \frac{d}{d x} [- 6]) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.5

Multiply $- 1$ by $1$ .

$\frac{(x - 3) (2 x - 1 + \frac{d}{d x} [- 6]) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.6

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{(x - 3) (2 x - 1 + 0) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.7

Add $2 x - 1$ and $0$ .

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 2.8

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

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 2.9

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

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6) (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 2.10

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

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6) (1 + 0)}{{(x - 3)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6) \cdot 1}{{(x - 3)}^{2}}$

Step 2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x - 3) (2 x - 1) - (x^{2} - x - 6)}{{(x - 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x - 3) (2 x - 1) - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x - 3) (2 x - 1)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 1) - 3 (2 x - 1) - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 1 - 3 (2 x - 1) - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.3

Move $- 1$ to the left of $x$ .

$\frac{2 x^{2} - 1 \cdot x - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.4

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} - x - 3 (2 x) - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.5

Multiply $2$ by $- 3$ .

$\frac{2 x^{2} - x - 6 x - 3 \cdot - 1 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.2.1.6

Multiply $- 3$ by $- 1$ .

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

Step 3.2.1.2.2

Subtract $6 x$ from $- x$ .

$\frac{2 x^{2} - 7 x + 3 - x^{2} - - x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.3

Multiply $- - x$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} - 7 x + 3 - x^{2} + 1 x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.3.2

Multiply $x$ by $1$ .

$\frac{2 x^{2} - 7 x + 3 - x^{2} + x - - 6}{{(x - 3)}^{2}}$

Step 3.2.1.4

Multiply $- 1$ by $- 6$ .

$\frac{2 x^{2} - 7 x + 3 - x^{2} + x + 6}{{(x - 3)}^{2}}$

Step 3.2.2

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

$\frac{x^{2} - 7 x + 3 + x + 6}{{(x - 3)}^{2}}$

Step 3.2.3

Add $- 7 x$ and $x$ .

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

Step 3.2.4

Add $3$ and $6$ .

$\frac{x^{2} - 6 x + 9}{{(x - 3)}^{2}}$

Step 3.3

Factor using the perfect square rule.

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Step 3.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 3.3.3

Rewrite the polynomial.

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

Step 3.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 3.4

Cancel the common factor of ${(x - 3)}^{2}$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

$1$