Find the derivative of the function.

Question

Find the derivative of the function.

y=e^8x-e^-8x/e^8x+e^-8x

Answer 1

I solve this 3 ways for you.

Brute force using quotient rule

[(8e^8x+8e^-8x)(e^8x+e^-8x)-(8e^8x-8e^-8x)(e^8x-e^-8x)]/(e^8x+e^-8x)^2 =

8[e^8x+e^-8x)^2-(e^8x-e^-8x)^2]/(e^8x+e^-8x)^2 = using that a^2-b^2=(a+b)(a-b)

8[(2e^8x)(2e^-8x)]/(e^8x+e^-8x)^2 =

32/(e^8x+e^-8x)^2

Two alternatives:

recognize that this is tanh(8x), whose derivative is 8sech^2(8x) or 32/(e^8x+e^-8x)^2

Rewrite as 1 - 2e^-8x/(e^8x+e^-8x) = 1 - 2/e^16x+1, whose derivative, now simple to calculate, is

32e^16x/(e^16x+1)^2 = 32/(e^8x+e^-8x)^2