Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d \cdot d} [\frac{f (d)}{g (d)}]$ is $\frac{g (d) \frac{d}{d \cdot d} [f (d)] - f (d) \frac{d}{d \cdot d} [g (d)]}{{g (d)}^{2}}$ where $f (d) = c x + d$ and $g (d) = a x + b$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{(a x + b) (0 + 1) - (c x + d) \frac{d}{d \cdot d} [a x + b]}{{(a x + b)}^{2}}$

Step 2.4

Add $0$ and $1$ .

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

Step 2.5

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

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

Step 2.6

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

$\frac{(a x + b) \cdot 1 - (c x + d) (0 + \frac{d}{d \cdot d} [b])}{{(a x + b)}^{2}}$

Step 2.7

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

$\frac{(a x + b) \cdot 1 - (c x + d) (0 + 0)}{{(a x + b)}^{2}}$

Step 2.8

Add $0$ and $0$ .

$\frac{(a x + b) \cdot 1 - (c x + d) \cdot 0}{{(a x + b)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{a x \cdot 1 + b \cdot 1 - (c x + d) \cdot 0}{{(a x + b)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{a x \cdot 1 + b \cdot 1 + (- (c x) - d) \cdot 0}{{(a x + b)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{a x \cdot 1 + b \cdot 1 - (c x) \cdot 0 - d \cdot 0}{{(a x + b)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $a$ by $1$ .

$\frac{a x + b \cdot 1 - (c x) \cdot 0 - d \cdot 0}{{(a x + b)}^{2}}$

Step 3.4.1.2

Multiply $b$ by $1$ .

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

Step 3.4.1.3

Multiply $- c x \cdot 0$ .

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Step 3.4.1.3.1

Multiply $0$ by $- 1$ .

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

Step 3.4.1.3.2

Multiply $0$ by $c$ .

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

Step 3.4.1.3.3

Multiply $0$ by $x$ .

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

Step 3.4.1.4

Multiply $- d \cdot 0$ .

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Step 3.4.1.4.1

Multiply $0$ by $- 1$ .

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

Step 3.4.1.4.2

Multiply $0$ by $d$ .

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

Step 3.4.2

Combine the opposite terms in $a x + b + 0 + 0$ .

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Step 3.4.2.1

Add $a x + b$ and $0$ .

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

Step 3.4.2.2

Add $a x + b$ and $0$ .

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

Step 3.5

Cancel the common factor of $a x + b$ and ${(a x + b)}^{2}$ .

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Step 3.5.1

Multiply by $1$ .

$\frac{(a x + b) \cdot 1}{{(a x + b)}^{2}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $a x + b$ out of ${(a x + b)}^{2}$ .

$\frac{(a x + b) \cdot 1}{(a x + b) (a x + b)}$

Step 3.5.2.2

Cancel the common factor.

$\frac{(a x + b) \cdot 1}{(a x + b) (a x + b)}$

Step 3.5.2.3

Rewrite the expression.

$\frac{1}{a x + b}$

Step 3.6

Reorder terms.

$\frac{1}{b + a x}$