Rate of Change of product of three functions

[tmlrlz]

Homework Statement

Find the rate of change of the product f(x)g(x)h(x) with respect to x at x =1 given that
f(1) = 0, g(1) = 2, h(1) = -2
f'(1) = 1, g'(1) = -1, h'(1) = 0

Homework Equations

f(x), g(x), h(x) (they are not provided)

The Attempt at a Solution

I decided to find an equation for f(x), g(x) and h(x) which satisfies all the above properties so after some guess work and a little bit of arithmetic i found that these equations will work given the conditions stated above:
f(x) = x-1
f'(x) = 1
g(x) = (x² - 4x + 7)/2
g'(x) = x-2
h(x) = -2
h'(x) = 0
Then,
f(x)g(x)h(x) = -2(x-1)[(x² - 4x + 7)/2)]
= (1-x)(x² - 4x + 7)
Derivative = -1(x² - 4x + 7) + (2x-4)(1-x)
= -x² + 4x - 7 - 2x² + 6x -4
= -3x² + 10x - 11
At x = 1
Derivative = -3 + 10 - 11
= -4
I'm just not sure if this is the right way to do it because even though my functions might work, there is no way of showing how i got them because they mostly just required some thinking and its not easy to explain how you come up with the process for finding the equations. Also, I'm pretty sure there are many other formulas that would have worked and many different combinations so I'm just wondering whether i did this question properly or if there is a better way to answer this question, maybe with something more general, I'm not sure, i just want some clarification on my answer. Thank you.

 

[Stephen Tashi]

Write out the derivative of f(x)g(x)h(x) symbolically by applying the product rule. You get:
f'(x)g(x)h(x) + f(x)g'(x)h(x) + ... etc. To evaluate this at x = 1, you get f'(1)g(1)h(1) + f(1)g'(1)h(1) + ... etc and you can find the numerical value of each term.

You should also get the correct answer if you invent functions that satisfy the given conditions, but it isn't necessary to deal with specific functions to do the problem.