# 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:

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:

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

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. $\frac{d}{dx}\left({5}^{x}+{e}^{x}\right)$ $\frac{d}{dx}{5}^{x}+\frac{d}{dx}{e}^{x}$ 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. CER $\frac{d}{dx}{e}^{x}={e}^{x}$          NER Step 4: Add/Subtract the derivatives & simplify.
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. $\frac{d}{dx}\left(6{e}^{x}+{x}^{2}-{12}^{x}\right)$ $\frac{d}{dx}6{e}^{x}+\frac{d}{dx}{x}^{2}-\frac{d}{dx}{12}^{x}$ 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. $\frac{d}{dx}6{e}^{x}=6\frac{d}{dx}{e}^{x}={6}{e}^{x}$ CM/NER $\frac{d}{dx}{x}^{2}=2{x}^{1}={2}{x}$ PR $\frac{d}{dx}{12}^{x}={\left(}{\mathrm{ln}}{12}{\right)}{12}^{x}$ CER Step 4: Add/Subtract the derivatives & simplify. $6{e}^{x}+2x-\left(\mathrm{ln}12\right){12}^{x}$
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. $\frac{d}{dx}\left(-4{e}^{x}+{10}^{x}\right)$ $\frac{d}{dx}-4{e}^{x}+\frac{d}{dx}{10}^{x}$ 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. $\frac{d}{dx}-4{e}^{x}=-4\frac{d}{dx}{e}^{x}={-}{4}{e}^{x}$ CM/NER $\frac{d}{dx}{10}^{x}={\left(}{\mathrm{ln}}{10}{\right)}{10}^{x}$      CER Step 4: Add/Subtract the derivatives & simplify. $-4{e}^{x}+\left(\mathrm{ln}10\right){10}^{x}$

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