Product rule with three functions, including functions f(x), g(x), and h(x), is a method in calculus that extends the product rule for two functions to situations with three functions. This technique allows for the computation of the derivative of the product of three functions, and it involves the multiplication of the first derivative of each function with the product of the other two functions.
The Product Rule for Three Functions
The product rule is a mathematical formula that allows us to differentiate the product of three functions. It is an extension of the product rule for two functions.
Formula:
(f(x)g(x)h(x))' = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)
Explanation:
The product rule for three functions involves three steps. Let’s consider functions f(x), g(x), and h(x).
- Differentiate the first function with respect to x: Calculate f'(x).
- Multiply the result by the product of the other two functions: Multiply f'(x) by g(x)h(x).
- Differentiate the second function with respect to x: Calculate g'(x).
- Multiply the result by the product of the other two functions: Multiply g'(x) by f(x)h(x).
- Differentiate the third function with respect to x: Calculate h'(x).
- Multiply the result by the product of the other two functions: Multiply h'(x) by f(x)g(x).
The sum of these three terms gives the derivative of the product of the three functions, (f(x)g(x)h(x))’.
Example:
Let’s find the derivative of the function (x^2) * (x+1) * (sin x).
- f(x) = x^2, g(x) = x+1, h(x) = sin x
- f'(x) = 2x
- g'(x) = 1
- h'(x) = cos x
Using the product rule:
((x^2) * (x+1) * (sin x))' = 2x * (x+1) * (sin x) + (x^2) * 1 * (sin x) + (x^2) * (x+1) * (cos x)
Simplifying:
(x^2 * (x+1) * sin x)' = 2x(x+1)sin x + x^2sinx + (x^2+x^3)cosx
Table Summary:
| Step | Operation | Result |
|---|---|---|
| 1 | Differentiate f(x) | f'(x) |
| 2 | Multiply by g(x)h(x) | f'(x)g(x)h(x) |
| 3 | Differentiate g(x) | g'(x) |
| 4 | Multiply by f(x)h(x) | f(x)g'(x)h(x) |
| 5 | Differentiate h(x) | h'(x) |
| 6 | Multiply by f(x)g(x) | f(x)g(x)h'(x) |
Question 1:
How to apply the product rule to differentiate a function involving three functions?
Answer:
To differentiate a function f(x) = g(x)h(x)i(x) using the product rule, multiply the first function g(x) by the derivative of the product of the second and third functions h(x)i(x), and then add the product of the derivative of g(x) with the product of the second and third functions h(x)i(x).
Question 2:
What is the key difference between the quotient rule and the product rule in differentiation?
Answer:
The product rule involves multiplying functions and differentiating the product, while the quotient rule involves dividing functions and differentiating the quotient.
Question 3:
How does the number of functions affect the complexity of using the product rule?
Answer:
The complexity of the product rule increases with the number of functions involved, as the number of terms in the derivative expression increases.
And there you have it, folks! We’ve taken a deep dive into the product rule with three functions, and hopefully, it all makes sense. If you’re still scratching your head, don’t worry—math is all about practice. So keep working at it, and eventually, it’ll click. Thanks for reading, and be sure to check back later for more math goodness!