Calculus Examples

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

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 $f (x) = a x + b$ and $g (x) = c$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $a$ by $1$ .

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

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

Step 4.2

Add $a$ and $0$ .

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

Step 4.3

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $- a x \cdot 0$ .

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Step 5.3.1.1.1

Multiply $0$ by $- 1$ .

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

Step 5.3.1.1.2

Multiply $0$ by $a$ .

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

Step 5.3.1.1.3

Multiply $0$ by $x$ .

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

Step 5.3.1.2

Multiply $- b \cdot 0$ .

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Step 5.3.1.2.1

Multiply $0$ by $- 1$ .

$\frac{c a + 0 + 0 b}{c^{2}}$

Step 5.3.1.2.2

Multiply $0$ by $b$ .

$\frac{c a + 0 + 0}{c^{2}}$

Step 5.3.2

Combine the opposite terms in $c a + 0 + 0$ .

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Step 5.3.2.1

Add $c a$ and $0$ .

$\frac{c a + 0}{c^{2}}$

Step 5.3.2.2

Add $c a$ and $0$ .

$\frac{c a}{c^{2}}$

Step 5.4

Cancel the common factor of $c$ and $c^{2}$ .

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Step 5.4.1

Factor $c$ out of $c a$ .

$\frac{c (a)}{c^{2}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $c$ out of $c^{2}$ .

$\frac{c (a)}{c \cdot c}$

Step 5.4.2.2

Cancel the common factor.

$\frac{c a}{c \cdot c}$

Step 5.4.2.3

Rewrite the expression.

$\frac{a}{c}$