Calculus Examples

$\frac{x - 5}{x^{2} - 3 x + 15}$

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 $f (x) = x - 5$ and $g (x) = x^{2} - 3 x + 15$ .

$\frac{(x^{2} - 3 x + 15) \frac{d}{d x} [x - 5] - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(x^{2} - 3 x + 15) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.2

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

$\frac{(x^{2} - 3 x + 15) (1 + \frac{d}{d x} [- 5]) - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.3

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

$\frac{(x^{2} - 3 x + 15) (1 + 0) - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} - 3 x + 15) \cdot 1 - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.4.2

Multiply $x^{2} - 3 x + 15$ by $1$ .

$\frac{x^{2} - 3 x + 15 - (x - 5) \frac{d}{d x} [x^{2} - 3 x + 15]}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $x^{2} - 3 x + 15$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [15]$ .

$\frac{x^{2} - 3 x + 15 - (x - 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [15])}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.6

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

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [15])}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.7

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

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [15])}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.8

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

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x - 3 \cdot 1 + \frac{d}{d x} [15])}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.9

Multiply $- 3$ by $1$ .

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x - 3 + \frac{d}{d x} [15])}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.10

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x - 3 + 0)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 2.11

Add $2 x - 3$ and $0$ .

$\frac{x^{2} - 3 x + 15 - (x - 5) (2 x - 3)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x^{2} - 3 x + 15 + (- x - - 5) (2 x - 3)}{{(x^{2} - 3 x + 15)}^{2}}$

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

Multiply $- 1$ by $- 5$ .

$\frac{x^{2} - 3 x + 15 + (- x + 5) (2 x - 3)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.2

Expand $(- x + 5) (2 x - 3)$ using the FOIL Method.

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Step 3.2.1.2.1

Apply the distributive property.

$\frac{x^{2} - 3 x + 15 - x (2 x - 3) + 5 (2 x - 3)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.2.2

Apply the distributive property.

$\frac{x^{2} - 3 x + 15 - x (2 x) - x \cdot - 3 + 5 (2 x - 3)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.2.3

Apply the distributive property.

$\frac{x^{2} - 3 x + 15 - x (2 x) - x \cdot - 3 + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3

Simplify and combine like terms.

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Step 3.2.1.3.1

Simplify each term.

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Step 3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 3 x + 15 - 1 \cdot 2 x \cdot x - x \cdot - 3 + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.2

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

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Step 3.2.1.3.1.2.1

Move $x$ .

$\frac{x^{2} - 3 x + 15 - 1 \cdot 2 (x \cdot x) - x \cdot - 3 + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 3 x + 15 - 1 \cdot 2 x^{2} - x \cdot - 3 + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.3

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 3 x + 15 - 2 x^{2} - x \cdot - 3 + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.4

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} - 3 x + 15 - 2 x^{2} + 3 x + 5 (2 x) + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.5

Multiply $2$ by $5$ .

$\frac{x^{2} - 3 x + 15 - 2 x^{2} + 3 x + 10 x + 5 \cdot - 3}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.1.6

Multiply $5$ by $- 3$ .

$\frac{x^{2} - 3 x + 15 - 2 x^{2} + 3 x + 10 x - 15}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.1.3.2

Add $3 x$ and $10 x$ .

$\frac{x^{2} - 3 x + 15 - 2 x^{2} + 13 x - 15}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.2

Combine the opposite terms in $x^{2} - 3 x + 15 - 2 x^{2} + 13 x - 15$ .

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Step 3.2.2.1

Subtract $15$ from $15$ .

$\frac{x^{2} - 3 x - 2 x^{2} + 13 x + 0}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.2.2

Add $x^{2} - 3 x - 2 x^{2} + 13 x$ and $0$ .

$\frac{x^{2} - 3 x - 2 x^{2} + 13 x}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.3

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

$\frac{- x^{2} - 3 x + 13 x}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.2.4

Add $- 3 x$ and $13 x$ .

$\frac{- x^{2} + 10 x}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.3

Factor $x$ out of $- x^{2} + 10 x$ .

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Step 3.3.1

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

$\frac{x (- x) + 10 x}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.3.2

Factor $x$ out of $10 x$ .

$\frac{x (- x) + x \cdot 10}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.3.3

Factor $x$ out of $x (- x) + x \cdot 10$ .

$\frac{x (- x + 10)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.4

Factor $- 1$ out of $- x$ .

$\frac{x (- (x) + 10)}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.5

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{x (- (x) - 1 (- 10))}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.6

Factor $- 1$ out of $- (x) - 1 (- 10)$ .

$\frac{x (- (x - 10))}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.7

Rewrite $- (x - 10)$ as $- 1 (x - 10)$ .

$\frac{x (- 1 (x - 10))}{{(x^{2} - 3 x + 15)}^{2}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{x (x - 10)}{{(x^{2} - 3 x + 15)}^{2}}$