Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $0$ and $0$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

Evaluate $\frac{d}{d b} [- b]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

Subtract $1$ from $0$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $a x \cdot 0$ .

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Step 5.4.1.1.1

Multiply $0$ by $a$ .

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

Step 5.4.1.1.2

Multiply $0$ by $x$ .

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

Step 5.4.1.2

Multiply $- b \cdot 0$ .

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Step 5.4.1.2.1

Multiply $0$ by $- 1$ .

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

Step 5.4.1.2.2

Multiply $0$ by $b$ .

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

Step 5.4.1.3

Multiply $- c x \cdot - 1$ .

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Step 5.4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.1.3.2

Multiply $c$ by $1$ .

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

Step 5.4.1.4

Multiply $- - d$ .

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Step 5.4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.1.4.2

Multiply $d$ by $1$ .

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

Step 5.4.1.5

Move $- 1$ to the left of $d$ .

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

Step 5.4.1.6

Rewrite $- 1 d$ as $- d$ .

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

Step 5.4.2

Combine the opposite terms in $0 + 0 + c x - d$ .

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Step 5.4.2.1

Add $0$ and $0$ .

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

Step 5.4.2.2

Add $0$ and $c x$ .

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

Step 5.5

Reorder terms.

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

Step 5.6

Factor $- 1$ out of $- d$ .

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

Step 5.7

Factor $- 1$ out of $c x$ .

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

Step 5.8

Factor $- 1$ out of $- (d) - (- c x)$ .

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

Step 5.9

Rewrite $- (d - c x)$ as $- 1 (d - c x)$ .

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

Step 5.10

Move the negative in front of the fraction.

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