Calculus Examples

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

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 $f (x) = q x + r$ and $g (x) = s x + t$ .

$\frac{(s x + t) \frac{d}{d x} [q x + r] - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(s x + t) (\frac{d}{d x} [q x] + \frac{d}{d x} [r]) - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.2

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

$\frac{(s x + t) (q \frac{d}{d x} [x] + \frac{d}{d x} [r]) - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.3

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

$\frac{(s x + t) (q \cdot 1 + \frac{d}{d x} [r]) - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.4

Multiply $q$ by $1$ .

$\frac{(s x + t) (q + \frac{d}{d x} [r]) - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.5

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

$\frac{(s x + t) (q + 0) - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.6

Add $q$ and $0$ .

$\frac{(s x + t) q - (q x + r) \frac{d}{d x} [s x + t]}{{(s x + t)}^{2}}$

Step 2.7

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

$\frac{(s x + t) q - (q x + r) (\frac{d}{d x} [s x] + \frac{d}{d x} [t])}{{(s x + t)}^{2}}$

Step 2.8

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

$\frac{(s x + t) q - (q x + r) (s \frac{d}{d x} [x] + \frac{d}{d x} [t])}{{(s x + t)}^{2}}$

Step 2.9

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

$\frac{(s x + t) q - (q x + r) (s \cdot 1 + \frac{d}{d x} [t])}{{(s x + t)}^{2}}$

Step 2.10

Multiply $s$ by $1$ .

$\frac{(s x + t) q - (q x + r) (s + \frac{d}{d x} [t])}{{(s x + t)}^{2}}$

Step 2.11

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

$\frac{(s x + t) q - (q x + r) (s + 0)}{{(s x + t)}^{2}}$

Step 2.12

Add $s$ and $0$ .

$\frac{(s x + t) q - (q x + r) s}{{(s x + t)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{s x q + t q - (q x + r) s}{{(s x + t)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{s x q + t q + (- (q x) - r) s}{{(s x + t)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{s x q + t q - (q x) s - r s}{{(s x + t)}^{2}}$

Step 3.4

Combine the opposite terms in $s x q + t q - (q x) s - r s$ .

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Step 3.4.1

Reorder the factors in the terms $s x q$ and $- (q x) s$ .

$\frac{q s x + t q - q s x - r s}{{(s x + t)}^{2}}$

Step 3.4.2

Subtract $q s x$ from $q s x$ .

$\frac{t q + 0 - r s}{{(s x + t)}^{2}}$

Step 3.4.3

Add $t q$ and $0$ .

$\frac{t q - r s}{{(s x + t)}^{2}}$

Step 3.5

Reorder terms.

$\frac{q t - r s}{{(s x + t)}^{2}}$