Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 44

f(t) = 3^(2t)/t
To take the derivative of this function, use the quotient rule (u/v)'= (v*u' - u*v')/v^2.
Applying that, f'(t) will be:
f'(t) = (t * (3^(2t))' - 3^(2t)*(t)')/t^2
f'(t) = (t*(3^(2t))' - 3^(2t) * 1)/t^2
Take note that the derivative formula of an exponential function is (a^u)' = ln(a) * a^u * u' .
So the derivative of 3^(2t) is:
f'(t) = (t*ln(3)*3^(2t) * (2t)' - 3^(2t) * 1)/t^2
f'(t)= (t*ln(3)*3^(2t) * 2 - 3^(2t) * 1)/t^2
f'(t)= (2t ln(3)*3^(2t) - 3^(2t))/t^2
f'(t) = (3^(2t)(2tln(3)-1))/t^2

Therefore, the derivative of the function is f'(t) = (3^(2t)(2tln(3)-1))/t^2 .