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

Answer 1

Charles Anctil

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

#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#

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Answer 2

Cameron Alexander

The derivative of ( \frac{{3^{2t}}}{t} ) with respect to ( t ) is ( \frac{{2 \cdot 3^{2t} \cdot (\ln(3) \cdot t - 1)}}{t^2} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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