Calculus Examples

$f (x) = \frac{a x + b}{c x + d}$

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 $f (x) = a x + b$ and $g (x) = c x + d$ .

$\frac{(c x + d) \frac{d}{d x} [a x + b] - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(c x + d) (\frac{d}{d x} [a x] + \frac{d}{d x} [b]) - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2.2

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

$\frac{(c x + d) (a \frac{d}{d x} [x] + \frac{d}{d x} [b]) - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2.3

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

$\frac{(c x + d) (a \cdot 1 + \frac{d}{d x} [b]) - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2.4

Multiply $a$ by $1$ .

$\frac{(c x + d) (a + \frac{d}{d x} [b]) - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2.5

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

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

Step 2.6

Add $a$ and $0$ .

$\frac{(c x + d) a - (a x + b) \frac{d}{d x} [c x + d]}{{(c x + d)}^{2}}$

Step 2.7

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

$\frac{(c x + d) a - (a x + b) (\frac{d}{d x} [c x] + \frac{d}{d x} [d])}{{(c x + d)}^{2}}$

Step 2.8

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

$\frac{(c x + d) a - (a x + b) (c \frac{d}{d x} [x] + \frac{d}{d x} [d])}{{(c x + d)}^{2}}$

Step 2.9

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

$\frac{(c x + d) a - (a x + b) (c \cdot 1 + \frac{d}{d x} [d])}{{(c x + d)}^{2}}$

Step 2.10

Multiply $c$ by $1$ .

$\frac{(c x + d) a - (a x + b) (c + \frac{d}{d x} [d])}{{(c x + d)}^{2}}$

Step 2.11

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

$\frac{(c x + d) a - (a x + b) (c + 0)}{{(c x + d)}^{2}}$

Step 2.12

Add $c$ and $0$ .

$\frac{(c x + d) a - (a x + b) c}{{(c x + d)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{c x a + d a - (a x + b) c}{{(c x + d)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{c x a + d a + (- (a x) - b) c}{{(c x + d)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{c x a + d a - (a x) c - b c}{{(c x + d)}^{2}}$

Step 3.4

Combine the opposite terms in $c x a + d a - (a x) c - b c$ .

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Step 3.4.1

Reorder the factors in the terms $c x a$ and $- (a x) c$ .

$\frac{a c x + d a - a c x - b c}{{(c x + d)}^{2}}$

Step 3.4.2

Subtract $a c x$ from $a c x$ .

$\frac{d a + 0 - b c}{{(c x + d)}^{2}}$

Step 3.4.3

Add $d a$ and $0$ .

$\frac{d a - b c}{{(c x + d)}^{2}}$

Step 3.5

Reorder terms.

$\frac{a d - b c}{{(d + c x)}^{2}}$