Calculus Examples

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

Step 1

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

$\frac{1}{c x + d} \frac{d}{d a} [a x + b]$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $x$ by $1$ .

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

Step 6

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

$\frac{1}{c x + d} (x + 0)$

Step 7

Combine fractions.

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Step 7.1

Add $x$ and $0$ .

$\frac{1}{c x + d} x$

Step 7.2

Combine $\frac{1}{c x + d}$ and $x$ .

$\frac{x}{c x + d}$

Step 7.3

Reorder terms.

$\frac{x}{d + c x}$