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How do you differentiate #f(x) = (tan(3x-2))/(e^(1-x)-1)# using the quotient rule?

Answer 1

Isaiah Allen

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Answer 2

Ezra Adams

To differentiate ( f(x) = \frac{\tan(3x - 2)}{e^{1-x} - 1} ) using the quotient rule, you can use the formula:

[ f'(x) = \frac{(g(x) \cdot f'(x) - f(x) \cdot g'(x))}{(g(x))^2} ]

where ( f(x) = \tan(3x - 2) ) and ( g(x) = e^{1-x} - 1 ).

First, find the derivatives of ( f(x) ) and ( g(x) ):

( f'(x) = 3 \sec^2(3x - 2) )

( g'(x) = -e^{1-x} )

Then, apply the quotient rule:

[ f'(x) = \frac{(e^{1-x} - 1) \cdot 3 \sec^2(3x - 2) - \tan(3x - 2) \cdot (-e^{1-x})}{(e^{1-x} - 1)^2} ]

Simplify this expression to get the final result.

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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.