Skill: 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

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

so

$$u'(x)=6x.$$

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

$$v(x)=x+4,$$

so

$$v'(x)=1.$$

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.