Calculus Examples

$\frac{x^{2} - 9}{x - 5}$

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 $f (x) = x^{2} - 9$ and $g (x) = x - 5$ .

$\frac{(x - 5) \frac{d}{d x} [x^{2} - 9] - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$ .

$\frac{(x - 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.2

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

$\frac{(x - 5) (2 x + \frac{d}{d x} [- 9]) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{(x - 5) (2 x + 0) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

$\frac{(x - 5) (2 x) - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.4.2

Move $2$ to the left of $x - 5$ .

$\frac{2 \cdot (x - 5) x - (x^{2} - 9) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{2 (x - 5) x - (x^{2} - 9) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 2.6

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

$\frac{2 (x - 5) x - (x^{2} - 9) (1 + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 2.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{2 (x - 5) x - (x^{2} - 9) (1 + 0)}{{(x - 5)}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

$\frac{2 (x - 5) x - (x^{2} - 9) \cdot 1}{{(x - 5)}^{2}}$

Step 2.8.2

Multiply $- 1$ by $1$ .

$\frac{2 (x - 5) x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 5) x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - (x^{2} - 9)}{{(x - 5)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

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 $x$ .

$\frac{2 (x \cdot x) + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 3.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 \cdot - 5 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 3.4.1.2

Multiply $2$ by $- 5$ .

$\frac{2 x^{2} - 10 x - x^{2} - - 9}{{(x - 5)}^{2}}$

Step 3.4.1.3

Multiply $- 1$ by $- 9$ .

$\frac{2 x^{2} - 10 x - x^{2} + 9}{{(x - 5)}^{2}}$

Step 3.4.2

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

$\frac{x^{2} - 10 x + 9}{{(x - 5)}^{2}}$

Step 3.5

Factor $x^{2} - 10 x + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 3.5.2

Write the factored form using these integers.

$\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$