Calculus Examples

$f (x) = \frac{q x + r}{s x + t}$

Step 1

Since $\frac{1}{s x + t}$ is constant with respect to $q$ , the derivative of $\frac{q x + r}{s x + t}$ with respect to $q$ is $\frac{1}{s x + t} \frac{d}{d q} [q x + r]$ .

$\frac{1}{s x + t} \frac{d}{d q} [q x + r]$

Step 2

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

$\frac{1}{s x + t} (\frac{d}{d q} [q x] + \frac{d}{d q} [r])$

Step 3

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

$\frac{1}{s x + t} (x \frac{d}{d q} [q] + \frac{d}{d q} [r])$

Step 4

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

$\frac{1}{s x + t} (x \cdot 1 + \frac{d}{d q} [r])$

Step 5

Multiply $x$ by $1$ .

$\frac{1}{s x + t} (x + \frac{d}{d q} [r])$

Step 6

Since $r$ is constant with respect to $q$ , the derivative of $r$ with respect to $q$ is $0$ .

$\frac{1}{s x + t} (x + 0)$

Step 7

Combine fractions.

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Step 7.1

Add $x$ and $0$ .

$\frac{1}{s x + t} x$

Step 7.2

Combine $\frac{1}{s x + t}$ and $x$ .

$\frac{x}{s x + t}$