Differentiate \frac{d}{dx}[\frac{1}{g(x)}] from first principles and use the result to derive the product rule assuming the product rule to be true.

I've differentiated it using the quotient rule (get \frac{-g'(x)}{(g(x))^2}) to use as a check and also by the chain rule but cannot reach the answer through first principles or derive the quotient rule using the answer I got for the first part by a different method.

Won't post all the workings, but I started with the definition of differentiation from first principles and let {f(x)=\frac{1}{g(x)} and worked through it but the closest I think I get is \frac{0}{(g(x))^2} and I think I even made errors with that.

So any help would be appreciated :)

Use the fact that \frac{1}{x} = x^{-1}. That should make it a form that is easy to differentiate.

Start with the definition of the derivative, like this:

\frac {d(1/g(x))} {dx} =\displaystyle\lim_{h \to 0} \( \frac { \frac 1 {(g(x+h)} - \frac 1 {g(x)}} h \)

Massage this a bit and you get:


- \displaystyle\lim_{h \to 0} \( \frac {g(x+h) - g(x)} h \) \cdot \frac 1 {g(x+h)g(x)}


Notice the first term is the definition of g'(x). Can you take it from here?

Capuchin - Tried and failed that way somehow

ebaines - Thanks, helps a bit.

Now managed to complete the whole question including deriving the quotient rule, many thanks again guys :thup: