General Question About Limits?

[alexedward]

I was asked to differentiate the function h(x) = g / (f^2+g), using the quotient law. I did so, and ended up with this:

h'(x) = [g'(f^2+g) - (2f' + g')(g)] / [(f^2+g)]^2.

When I asked a T.A if this was correct, they said no. According to them, the part in the numerator that reads (2f' + g')(g) should actually be (2ff' + g')(g) (there is a prime symbol beside the second f, and the first g). I know what the T.A is telling me is correct, since it got me the right answer..but why do you do this? Is this different from the derivative of x^2, which is 2x?

 

[mmm4444bot]

Looks like the Chain Rule, to me.

What do the symbols f and g represent ? Did they tell you that f and g are both functions of x ?

It's a recursive definition to define f(x) in terms of f(x) and g(x). :?

 

[alexedward]

Yes, they are both functions of x. Sorry for the notation, my main question was just to ask why, when doing the chain rule, f^2 becomes 2ff'. This may be an entirely stupid question, I'm just really confused right now.

 

[mmm4444bot]

It appears that f is a composite function; it's defined by itself and another function of x called g.

When we derivate a composite function, we need to use the Chain Rule.

f(x) = g(x)/[f(x)^2 + g(x)]

It's sorta like g(x) is the inner function, but I do not understand defining f(x) in terms of itself. When I try to think of the derivative using limits, the recursion confuses me too much.

Perhaps, the exercise is meant as a thought experiment.