Rate of Change of product of three functions

In summary, to find the rate of change of the product f(x)g(x)h(x) with respect to x at x=1, you can use the product rule to find the derivative symbolically and then evaluate at x=1 to get the numerical value. While it is possible to find specific functions that satisfy the given conditions, it is not necessary to do so to solve the problem.
  • #1
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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.
 
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  • #2
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.
 

1. What is the rate of change of a product of three functions?

The rate of change of a product of three functions is the rate at which the product of the three functions is changing with respect to a given variable. It can be calculated by taking the derivative of the product of the three functions with respect to the variable of interest.

2. How do you find the rate of change of a product of three functions?

To find the rate of change of a product of three functions, you can use the product rule of differentiation. This rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. By applying this rule multiple times, you can find the derivative of a product of three functions.

3. Why is the rate of change of a product of three functions important?

The rate of change of a product of three functions is important because it helps in understanding how the product of the three functions is changing. It can be used to analyze and predict the behavior of complex systems that involve multiple functions. It is also a fundamental concept in calculus and is used in various real-world applications, such as physics, economics, and engineering.

4. Can the rate of change of a product of three functions be negative?

Yes, the rate of change of a product of three functions can be negative. This means that the product of the three functions is decreasing with respect to the given variable. It is important to note that the rate of change can be positive, negative, or zero depending on the values of the functions and the variable of interest.

5. How can the rate of change of a product of three functions be applied in real life?

The rate of change of a product of three functions can be applied in various real-life situations, such as in economics for analyzing the relationship between different variables, in physics for understanding the motion of objects, and in engineering for designing and optimizing systems. It can also be used in analyzing and predicting trends in business and financial markets.

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