Derivatives Optimization problem step.

Question

I don't understand a step in a calculus problem I'm working on. It goes something like this...

E(t) = 5 + 0.5te^-t/20

Then from there that simplifies into...

E'(t) = 0 + 0.5e^-t/20 + 0.5te^-t/20 (-1/20)

To me this looks sorta like product rule since the derivative of 5 would be 0, but I don't understand why the t would vanish in the first half of the equation or where the (-1/20) comes from.

Answer 1

When you take the derivative of e^u it is e^u du

du is the derivative of the exponent of e

So the derivative of e^-t/20 is e^-t/20 (-1/20),

To do the whole derivative we take the first times the derivative of the second plus the second times the derivative of the first

The first is 0.5t times derivative of second from above e^-t/20 (-1/20) to get 0.5te^-t/20 (-1/20),

then add to that the second e^-t/20 times the derivative of the first

the first is 0.5t the derivative of that is 0.5 so

The second times derivative of the first is 0.5e^-t/20

Putting all together

E'(t) = 0.5e^-t/20 + 0.5te^-t/20 (-1/20)