Week 1 quiz lesson 2

Ok so I have a question, if f(x) is differentiable at all points how is log f(x) differentiable at all points because log f(x) is not differentiable for any negative x values.

Hey @Rehan_G,
Welcome, and we are glad that you could become a part of our community :partying_face:

I believe you are referring to the 8th question in the quiz. Note that in the question, it is specified that “Let f(x) be a positive real function”, i.e., log(f(x)) will always exist and will always be differentiable, as long as f(x) is differentiable.

Now, let’s ignore the quiz once, and get back to this statement. First of all, x can be negative as long as f(x) is positive, because note that f(x) is the input to the log function, and not x. Second, whenever we apply log to any variable or function, we always impose the limit (implicitly or explicitly) that the input to log must be greater than 0, since log(x), for x <= 0 doesn’t exist. So, the aspect of differentiability never arises here, since the function doesn’t even exist.

That’s why, in the quiz, it has been explicitly mentioned that f(x) is a positive real function, i.e., f(x) > 0. Let us know if this resolves your issue.

Cheers,
Elemento