`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` .
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