# Thread: Derivatives of Exponential Functions

1. ## Derivatives of Exponential Functions

Hi Guys, can you explain a simple rule when finding the derivatives on exponential functions please?

Find the Deriviative:

g(x)=x^2e^4x^3

Use the product rule, then simplfy.

g'(x)=(x^2)'(e^4x^3) + (x^2)(e^4x^3)'
g'(x)= (2x)(e^4x^3) + (x^2)(e^4x^3)(12x^2)
g'(x)=2xe^4x^3+12x^4e^4x^3

I understand this completely, except for the 12X^2 in the second step. In this situation, why does the derivative of the exponential function e^4x^3 become part of the base and not e^12x^2?

I think this may solve the many problems I'm having with natural logarithms and exponential functions, staring at the function doesn't work anymore

thanks!

2. Originally Posted by Butum
Hi Guys, can you explain a simple rule when finding the derivatives on exponential functions please?

Find the Deriviative:

g(x)=x^2e^4x^3

Use the product rule, then simplfy.

g'(x)=(x^2)'(e^4x^3) + (x^2)(e^4x^3)'
g'(x)= (2x)(e^4x^3) + (x^2)(e^4x^3)(12x^2)
g'(x)=2xe^4x^3+12x^4e^4x^3

I understand this completely, except for the 12X^2 in the second step. In this situation, why does the derivative of the exponential function e^4x^3 become part of the base and not e^12x^2?

I think this may solve the many problems I'm having with natural logarithms and exponential functions, staring at the function doesn't work anymore

thanks!
The 12x^2 comes from the chain rule.

In general, the derivative of

$e^{f(x)}$

is

$e^{f(x)} f'(x)$.

For more arbitrary base:

$a^{h(x)}$

derives to

$\ln a a^{h(x)} h'(x)$

Example:

The derivative of

$2^{\cos x}$

is

$\ln{2}\text{ }2^{\cos x} (-\sin x)$

Good luck!

3. So we cannot fully use the chain rule when Euler's constant is a base? Is this because Euler's Constant is not considered a positive constant?

So for example, we will never see:

y=e^-(2x+5)
y'=e^-(2x+5)Ln e(-2)

I apologize if this is breaking the forum rule for posting two questions within one topic.

4. The chain rule will always be applied as long as there is a composition of functions.

Remember that ln e = 1, so we don't write it.

5. That is so obvious to me now, thank you so much.

I'll use the chain rule and just clear the Ln e when simplifying so I do not confuse myself.