Question

What is the derivative of `(3^(2t))/t`?

Answer 1

`f'(t) = ((2ln3*t-1)*3^(2t))/t^2`

Explanation:

`f(t)=3^(2t)/t = 3^(2t)*t^-1`

`ln f(t) = 2t*ln3-lnt`

Using implicit differentiation:
`1/f(t)*f'(t) = 2t*0 + 2*ln3 -1/t` (Product rule and standard differential)

`1/f(t)*f'(t) = 2ln3-1/t`

Since `f(t)=3^(2t)/t`
`f'(t) = 3^(2t)/t * (2ln3-1/t)`

`= = ((2ln3*t-1)*3^(2t))/t^2`