Calculus Examples

$f (x) = \frac{x + 5}{x + 4}$

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 $g (x) = x + 4$ .

$\frac{(x + 4) \frac{d}{d x} [x + 5] - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2.2

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

$\frac{(x + 4) (1 + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2.3

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

$\frac{(x + 4) (1 + 0) - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$\frac{(x + 4) \cdot 1 - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2.4.2

Multiply $x + 4$ by $1$ .

$\frac{x + 4 - (x + 5) \frac{d}{d x} [x + 4]}{{(x + 4)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$\frac{x + 4 - (x + 5) (\frac{d}{d x} [x] + \frac{d}{d x} [4])}{{(x + 4)}^{2}}$

Step 2.6

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

$\frac{x + 4 - (x + 5) (1 + \frac{d}{d x} [4])}{{(x + 4)}^{2}}$

Step 2.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x + 4 - (x + 5) (1 + 0)}{{(x + 4)}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

$\frac{x + 4 - (x + 5) \cdot 1}{{(x + 4)}^{2}}$

Step 2.8.2

Multiply $- 1$ by $1$ .

$\frac{x + 4 - (x + 5)}{{(x + 4)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x + 4 - x - 1 \cdot 5}{{(x + 4)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Combine the opposite terms in $x + 4 - x - 1 \cdot 5$ .

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Step 3.2.1.1

Subtract $x$ from $x$ .

$\frac{0 + 4 - 1 \cdot 5}{{(x + 4)}^{2}}$

Step 3.2.1.2

Add $0$ and $4$ .

$\frac{4 - 1 \cdot 5}{{(x + 4)}^{2}}$

Step 3.2.2

Multiply $- 1$ by $5$ .

$\frac{4 - 5}{{(x + 4)}^{2}}$

Step 3.2.3

Subtract $5$ from $4$ .

$\frac{- 1}{{(x + 4)}^{2}}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{1}{{(x + 4)}^{2}}$