Differentiate exponential functions

Differentiation of exponential functions establishes that $\frac{d}{dx}e^x=e^x$ and, for a positive constant base $a\ne1$ , $\frac{d}{dx}a^x=a^x\ln a$ , interpreting the derivative as an instantaneous rate proportional to the function’s current value. The chain rule extends these relationships to expressions such as $e^{u(x)}$ and $a^{u(x)}$ , while distinguishing the special role of base $e$ from other bases; variable-base exponentials and more advanced generalizations are not included.

Detailed Explanation: Differentiate exponential functions

For exponential functions, use these basic rules:

\frac{d}{dx}e^x=e^x

For a positive constant base $a\ne 1$ ,

\frac{d}{dx}a^x=a^x\ln(a).

If the exponent is a function $u(x)$ , use the chain rule:

\frac{d}{dx}e^{u(x)}=e^{u(x)}u'(x)

and

\frac{d}{dx}a^{u(x)}=a^{u(x)}\ln(a)\,u'(x).

The factor $\ln(a)$ is needed for bases other than $e$ . The derivative is proportional to the function’s current value.

Example

Differentiate

f(x)=e^{3x^2-1}+2^{x+4}.

Step 1: Differentiate the first term.

The exponent is

u(x)=3x^2-1,

u'(x)=6x.

Using the chain rule,

\frac{d}{dx}e^{3x^2-1} =e^{3x^2-1}(6x).

There is no $\ln(e)$ factor because $\ln(e)=1$ .

Step 2: Differentiate the second term.

The exponent is

v(x)=x+4,

v'(x)=1.

Since the base is $2$ , include the factor $\ln(2)$ :

\frac{d}{dx}2^{x+4} =2^{x+4}\ln(2)(1) =2^{x+4}\ln(2).

Step 3: Combine the results.

Therefore,

\boxed{f'(x)=6x e^{3x^2-1}+2^{x+4}\ln(2)}.

For each exponential term, keep the original exponential, multiply by $\ln(\text{base})$ when the base is not $e$ , and then multiply by the derivative of the exponent.

Learn by doing: Differentiate exponential functions

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Derivative Rules - General Exponential Exponent with Power to Derivative

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