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!