product rule problem example

Answer 1

Here's an example:

Let's say we have the function f(x) = (3x^2 + 2x)(4x^3 - 5x).

To differentiate this function using the product rule, we follow these steps:

1. Identify the two functions being multiplied together: (3x^2 + 2x) and (4x^3 - 5x).

2. Differentiate the first function with respect to x:
- The derivative of 3x^2 + 2x with respect to x is 6x + 2.

3. Keep the second function as it is.

4. Differentiate the second function with respect to x:
- The derivative of 4x^3 - 5x with respect to x is 12x^2 - 5.

5. Apply the product rule formula:
- The derivative of the product of two functions is given by (first function derivative)(second function) + (first function)(second function derivative).

So, applying the product rule to the given function, we have:

f'(x) = (6x + 2)(4x^3 - 5x) + (3x^2 + 2x)(12x^2 - 5).

Simplifying further if needed, but this is the general form of the derivative using the product rule for the given function.