Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d c} [c x]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $x$ by $1$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Add $x$ and $0$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $0$ and $0$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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 $x$ by $x$ by adding the exponents.

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Step 5.4.1.1.1

Move $x$ .

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

Step 5.4.1.1.2

Multiply $x$ by $x$ .

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

Step 5.4.1.2

Multiply $- c x \cdot 0$ .

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Step 5.4.1.2.1

Multiply $0$ by $- 1$ .

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

Step 5.4.1.2.2

Multiply $0$ by $c$ .

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

Step 5.4.1.2.3

Multiply $0$ by $x$ .

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

Step 5.4.1.3

Multiply $- a \cdot 0$ .

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Step 5.4.1.3.1

Multiply $0$ by $- 1$ .

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

Step 5.4.1.3.2

Multiply $0$ by $a$ .

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

Step 5.4.2

Combine the opposite terms in $a x^{2} + b x + 0 + 0$ .

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Step 5.4.2.1

Add $a x^{2} + b x$ and $0$ .

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

Step 5.4.2.2

Add $a x^{2} + b x$ and $0$ .

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

Step 5.5

Reorder terms.

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

Step 5.6

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

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Step 5.6.1

Factor $x$ out of $x^{2} a$ .

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

Step 5.6.2

Factor $x$ out of $b x$ .

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

Step 5.6.3

Factor $x$ out of $x (x a) + x b$ .

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

Step 5.7

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

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Step 5.7.1

Reorder terms.

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

Step 5.7.2

Factor $b + a x$ out of $x (b + a x)$ .

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

Step 5.7.3

Cancel the common factors.

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Step 5.7.3.1

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

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

Step 5.7.3.2

Cancel the common factor.

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

Step 5.7.3.3

Rewrite the expression.

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