Exponential Quotient (solve without calculator)

Question

Given y=(2^-2x)/(e^-2x), find dy/dx.

Answer 1

Rewrite as y = (2/e)^-2x

dy/dx = -2 ln(2/e) (2/e)^-2x

If you want to remove the negatives we can do that and make it a bit more neat:

dy/dx = -2 ln(2/e) (e/2)^2x

dy/dx = 2 ln(e/2) (e/2)^2x

Finally you could write ln(e/2) as ln e - ln 2

dy/dx = 2 [ln e - ln 2](e/2)^2x

dy/dx = 2 [1 - ln 2](e/2)^2x

Here is an exercise for you! Try to do it with the quotient rule and verify that you get the same result.

Answer 2

Video Attached!

See the attached video for my solution. I made sure to address Patrick F's challenge at the end of his solution: "Here is an exercise for you! Try to do it with the quotient rule and verify that you get the same result."