Question

How do I find the derivative of `3e^(-12t)`?

Answer 1

You can use the chain rule.

`(3e^(-12t))'=-36*e^(-12t)`

Explanation:

The 3 is a constant, it can be kept out:

`(3e^(-12t))'=3(e^(-12t))'`

It's a mixed function. The outer function is the exponential, and the inner is a polynomial (sort of):

`3(e^(-12t))'=3*e^(-12t)*(-12t)'=`

`=3*e^(-12t)*(-12)=-36*e^(-12t)`

Deriving:

If the exponent was a simple variable and not a function, we would simply differentiate `e^x`. However, the exponent is a function and should be transformed. Let `(3e^(-12t))=y` and `-12t=z`, then the derivative is:

`(dy)/dt=(dy)/dt*(dz)/dz=(dy)/dz*(dz)/dt`

Which means you differentiate `e^(-12t)` as if it were `e^x` (unchanged), then you differentiate `z` which is `-12t` and finally you multiply them.