Derivative of Composition Functions Problem

[Loppyfoot]

Homework Statement

f and g are differentiable functions that have the following properties:
i. f(x) < 0 for all values of x
ii. g(5) = 2

If h(x)= f(x) / g(x) and h'(x) = f '(x) / g(x), then g(x) = _____?

Homework Equations

Quotient Rule with f(x) and g(x)

The Attempt at a Solution

I have no idea where to begin on this problem.
Would I first solve the derivative of h(x)?

so h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))²
then set that equal to the second given (h'(x))= f'(x)/g(x)?

Would that be a correct way to start?

 

[lanedance]

if you mean equate the two expressions
h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))^2 = f '(x)*g(x)

then yes, sounds good

 

[Loppyfoot]

From there,I have no idea where to go next. would the answer to g(x) be an actual number of a function?

 

[Mark44]

Unless g(x) is a constant function, its formula will never be just a number. g(2), for example, would be a number, but g(x) will be a formula that gives the output for an arbitrary input number x.

 

[lanedance]

so if h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))^2 = f '(x)*g(x)
then

h'(x) - f '(x)*g(x) = - f(x)*g'(x) / (g(x))^2 = 0

so the term f(x)*g'(x) / (g(x))^2 is zero for all x, you know f(x) <0 for all x, so this is always non-zero, g(5)=2 so this is non-zero for at least one x, what does this tell you about g'(x)