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:


d dx ( a x )=( ln 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, ex, is equal to ex.

DERIVATIVE OF NATURAL EXPONENTIAL FUNCTION:


d dx ( e x )= e x



Let's take a look at a couple of examples

5x + ex

Step 1: Simplify the expression


This expression is already simplified.

5x + ex

Step 2: Apply the sum/difference rules.


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

d dx ( 5 x + e x )


d dx 5 x + d dx e x

Step 3: Take the derivative of each part.


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


Use the natural exponential rule (NER) to differentiate ex.

d dx 5 x =( ln 5 ) 5 x CER


d dx e x = e x          NER

Step 4: Add/Subtract the derivatives & simplify.

( ln 5 ) 5 x + e x

Example 1:     6ex + x2 - 12x

Step 1: Simplify the expression


This expression is already simplified.

6ex + x2 - 12x

Step 2: Apply the sum/difference rules.


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

d dx ( 6 e x + x 2 12 x )


d dx 6 e x + d dx x 2 d dx 12 x

Step 3: Take the derivative of each part.


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


Use the power rule (PR) to differentiate x2.


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

d dx 6 e x =6 d dx e x =6 e x CM/NER


d dx x 2 =2 x 1 =2x PR


d dx 12 x =(ln12) 12 x CER

Step 4: Add/Subtract the derivatives & simplify.

6 e x +2x(ln12) 12 x

Example 2:     -4ex + 10x

Step 1: Simplify the expression


This expression is already simplified.

-4ex + 10x

Step 2: Apply the sum/difference rules.


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

d dx ( 4 e x + 10 x )


d dx 4 e x + d dx 10 x

Step 3: Take the derivative of each part.


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


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

d dx 4 e x =4 d dx e x =4 e x CM/NER


d dx 10 x =(ln10) 10 x      CER

Step 4: Add/Subtract the derivatives & simplify.

4 e x +(ln10) 10 x





Related Links:
Math
algebra
Product Rule
Quotient Rule
Calculus Topics


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