Exponential Differentiation Rules

The general differentiation rules discussed how to differentiate a single function. This discussion will focus on how to differentiate polynomial functions involving addition and subtraction.

When functions are added together, the Sum Rule of Differentiation is used. This rule states that the derivative of the sum of the functions is equal to the sum of the derivatives of each part.

SUM RULE OF DIFFERENTIATION:

$\frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)$

When functions are subtracted, the Difference Rule of Differentiation is used. This rule states that the derivative of the difference of the functions is equal to the difference of the derivative of the parts.

DIFFERENCE RULE OF DIFFERENTIATION:

$\frac{d}{d x} [f (x) - g (x)] = \frac{d}{d x} f (x) - \frac{d}{d x} g (x)$

Let's take a look at a couple of examples

x³ - 7x + 12

Step 1: Simplify the expression

This expression is already simplified.

x³ - 7x + 12

Step 2: Apply the sum/difference rules.

Rewrite the derivative of the function as the sum/difference of the derivative of the parts.

$\frac{d}{d x} (x^{3} - 7 x + 12)$

$\frac{d}{d x} x^{3} - \frac{d}{d x} 7 x + \frac{d}{d x} 12$

Step 3: Take the derivative of each part.

Use the power rule to differentiate x³.

Use the constant multiple and power rules to differentiate 7x.

Use the constant rule to differentiate 12.

$\frac{d}{d x} x^{3} = 3 x^{2}$ Power Rule

$\frac{d}{d x} 7 x = 7 \frac{d}{d x} x = 7 x^{0} = 7$ CM & Power

$\frac{d}{d x} 12 = 0$ Constant Rule

Step 4: Add/Subtract the derivatives & simplify.

3x² - 7

Example 1: 2x³(3 - x)

Step 1: Simplify the expression

Distribute the 2x³.

6x³ - 2x⁴

Step 2: Apply the sum/difference rules.

Rewrite the derivative of the function as the sum/difference of the derivative of the parts.

$\frac{d}{d x} (6 x^{3} - 2 x^{4})$

$\frac{d}{d x} 6 x^{3} - \frac{d}{d x} 2 x^{4}$

Step 3: Take the derivative of each part.

Use the constant multiple and power rules to differentiate 6x³.

Use the constant multiple and power rules to differentiate 2x⁴.

$\frac{d}{d x} 6 x^{3} = 6 \frac{d}{d x} x^{3} = 18 x^{2}$ CM & Power

$\frac{d}{d x} 2 x^{4} = 2 \frac{d}{d x} x^{4} = 8 x^{3}$ CM & Power

Step 4: Add/Subtract the derivatives & simplify.

18x² - 8x³

Example 2:

(x^{6} + 3 x^{4} - x^{3}) + (\frac{1}{3 x^{2}} - 1)

Step 1: Simplify the expression

Rewrite $\frac{1}{3 x^{2}}$ as $\frac{1}{3} x^{- 2}$ and remove the parentheses.

$\frac{d}{d x} [(x^{6} + 3 x^{4} - x^{3}) + (\frac{1}{3 x^{2}} - 1)]$

$\frac{d}{d x} [x^{6} + 3 x^{4} - x^{3} + \frac{1}{3} x^{- 2} - 1]$

Step 2: Apply the sum/difference rules.

Rewrite the derivative of the function and the sum/difference of the derivatives of the parts.

$\frac{d}{d x} x^{6} + \frac{d}{d x} 3 x^{4} - \frac{d}{d x} x^{3} + \frac{d}{d x} \frac{1}{3} x^{- 2} - \frac{d}{d x} 1$

Step 3: Take the derivative of each part.

Use the power rule to differentiate x⁶

Use the constant multiple and power rules to differentiate 3x⁴.

Use the power rule to differentiate x³

Use the constant multiple and power rules to differentiate $\frac{1}{3} x^{- 2}$

Use the constant rule to differentiate 1

$\frac{d}{d x} x^{6} = 6 x^{5}$ Power

$\frac{d}{d x} 3 x^{4} = 3 \frac{d}{d x} x^{4} = 12 x^{3}$ CM & Power

$\frac{d}{d x} x^{3} = 3 x^{2}$ Power

$\frac{d}{d x} \frac{1}{3} x^{- 2} = \frac{1}{3} \frac{d}{d x} x^{- 2} = - \frac{2}{3} x^{- 3}$ CMPower

$\frac{d}{d x} 1 = 0$ Constant

Step 4: Add/Subtract the derivatives & simplify.

$6 x^{5} + 12 x^{3} - 3 x^{2} - \frac{2}{3} x^{- 3}$

Related Links:
Math
algebra
Common Base Exponential Differentiation Rules
Product Rule
Algebra Topics

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