Question

In: Advanced Math

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b). Assume...

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b).

Assume that g(x) dne 0 for all x ∈ (a, b). Prove that f/g is differentiable and (f/g)'(x) = (f'(x)g(x)-f(x)g'(x))/(g^2(x))

for all x ∈ (a, b)

Expert Solution

Let, and

Since, is differentiable at , is continuous at .

Since

There exists a neighbourhood of such that , for all

therefore for ,

Then,

since,

is continuous at

since and are differentiable at ,

and

therefore

and this shows that is differentiable at ,which is any point arbitrarily choosen from the given interval and

Colby Messinger answered 4 months ago

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