How do I combine the product rule and the chain rule in the simplest way possible?

To combine the product rule and the chain rule in the simplest way possible, follow these steps:

1. Identify the function that you need to differentiate. Let's call it f(x).

2. Determine whether the function can be expressed as the product of two simpler functions. If it can, write it in the form f(x) = g(x) * h(x), where g(x) and h(x) are the two simpler functions.

3. Apply the product rule by taking the derivative of f(x) with respect to x. The product rule states that d/dx(g(x) * h(x)) = g'(x) * h(x) + g(x) * h'(x).

4. Apply the chain rule to differentiate each of the simpler functions g(x) and h(x). The chain rule states that if h(x) is a function of g(x) and g(x) is a function of x, then d/dx(h(g(x))) = h'(g(x)) * g'(x).

5. Substitute the derivatives obtained from applying the chain rule into the product rule formula. This will give you the final expression for the derivative of f(x).

By following these steps, you will be able to combine the product rule and the chain rule to find the derivative of a function expressed as the product of two simpler functions.