Common Base Exponential Differentiation Rules

There are two basic differentiation rules for exponential equations.

The first rule is for Common Base Exponential Function, where a is any constant. To obtain the derivative take the natural log of the base (a) and multiply it by the exponent.

DERIVATIVE OF COMMON EXPONENTIAL FUNCTION:

$\frac{d}{d x} (a^{x}) = (l n a) a^{x}$

The second rule is for the natural exponential function, when a = e, where e is the irrational number approximated as 2.718. The derivative of the Natural Exponential Function, e^x, is equal to e^x.

DERIVATIVE OF NATURAL EXPONENTIAL FUNCTION:

$\frac{d}{d x} (e^{x}) = e^{x}$

Let's take a look at a couple of examples

5^x + e^x

Step 1: Simplify the expression

This expression is already simplified.

5^x + e^x

Step 2: Apply the sum/difference rules.

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

$\frac{d}{d x} (5^{x} + e^{x})$

$\frac{d}{d x} 5^{x} + \frac{d}{d x} e^{x}$

Step 3: Take the derivative of each part.

Use the common exponential rule (CER) to differentiate 5^x.

Use the natural exponential rule (NER) to differentiate e^x.

$\frac{d}{d x} 5^{x} = (l n 5) 5^{x}$ CER

$\frac{d}{d x} e^{x} = e^{x}$ NER

Step 4: Add/Subtract the derivatives & simplify.

$(l n 5) 5^{x} + e^{x}$

Example 1: 6e^x + x² - 12^x

Step 1: Simplify the expression

This expression is already simplified.

6e^x + x² - 12^x

Step 2: Apply the sum/difference rules.

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

$\frac{d}{d x} (6 e^{x} + x^{2} - 12^{x})$

$\frac{d}{d x} 6 e^{x} + \frac{d}{d x} x^{2} - \frac{d}{d x} 12^{x}$

Step 3: Take the derivative of each part.

Use the constant multiple and natural exponential rules (CM/NER) to differentiate 6e^x.

Use the power rule (PR) to differentiate x².

Use the common exponential rule (CER) to differentiate 12^x.

$\frac{d}{d x} 6 e^{x} = 6 \frac{d}{d x} e^{x} = 6 e^{x}$ CM/NER

$\frac{d}{d x} x^{2} = 2 x^{1} = 2 x$ PR

$\frac{d}{d x} 12^{x} = (\ln 12) 12^{x}$ CER

Step 4: Add/Subtract the derivatives & simplify.

$6 e^{x} + 2 x - (\ln 12) 12^{x}$

Example 2: -4e^x + 10^x

Step 1: Simplify the expression

This expression is already simplified.

-4e^x + 10^x

Step 2: Apply the sum/difference rules.

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

$\frac{d}{d x} (- 4 e^{x} + 10^{x})$

$\frac{d}{d x} - 4 e^{x} + \frac{d}{d x} 10^{x}$

Step 3: Take the derivative of each part.

Use the constant multiple and natural exponential rules (CM/NER) to differentiate -4e^x.

Use the common exponential rule (CER) to differentiate 10^x.

$\frac{d}{d x} - 4 e^{x} = - 4 \frac{d}{d x} e^{x} = - 4 e^{x}$ CM/NER

$\frac{d}{d x} 10^{x} = (\ln 10) 10^{x}$ CER

Step 4: Add/Subtract the derivatives & simplify.

$- 4 e^{x} + (\ln 10) 10^{x}$

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