Calculus question: derivatives

[Dantes]

If I get a question that gives me a function defined in variables and then certain F(X) and F'(X) with numbers that equal the slopes, and I need to find the derivative of the first function do I just substitute in.

For example:

Find the value of F'(8) when

f(x)/f(x)-g(x)

while f(8)= 3, f'(8) = 2 , g(8) = 4 , g'(8) = 3

do I just do:

F(X) = 3 / 3-4 and then use Newtonian qoutient to got the derivative of F(X) at 8 ?

[maverick280857]

The question you have asked Dantes is part of functional equations. Normally, one tries to use all the data given in a problem and ingeniously get to an equation/relationship that was sought at the start.

According to your problem, you have to find the derivative of F at the point x = 8.

First off, do you mean that

F(x) = f(x)/f(x)-g(x)
?

Secondly, how do you think you can find the derivative at a point after substituting the value of the independent variable in the function? Essentially if you have to find f'(c)--the derivative of f(x) at a point c--you would (even by first principles) find first f'(x) at a general point x in the domain of f(x) where f(x) is derivable. Then, you would substitute x = c to get f'(c). However, if you find f(c) first, and differentiate, you get a zero as f(c) is a constant value. This point, however has nothing explicit to do with your problem except the last line of your post as I comprehend it.

Let's say you have

F(x) = f(x)/f(x)-g(x)

Then for all f(x) not equal to zero

F(x) = 1-g(x)

so that

\frac{dF}{dx} = -\frac{dg}{dx} = -g'(x)

Cheers
Vivek