How do you find the first and second derivatives of #f(x)= (e^x)/(ln(x)+1)# using the quotient rule?

Answer 1

Samantha Adkison

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Answer 2

Christopher Agresta

To find the first derivative of ( f(x) = \frac{e^x}{\ln(x) + 1} ) using the quotient rule, follow these steps:

Let ( u(x) = e^x ) and ( v(x) = \ln(x) + 1 ).
Apply the quotient rule: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ).
Compute ( u'(x) ) and ( v'(x) ).
Substitute the values into the quotient rule formula.
Simplify the expression if possible.

For the second derivative, apply the quotient rule again to the first derivative function obtained.

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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.