I have to find the derivative of the following function:
$$f(x) = (x^3+ 4)(4x^5 + 2x − 5)^{1/2}$$
To start solving this, I've dissected the equation and realize that I must use the product and chain rule to find the derivative. The problem is I don't know how to use both at the same time. How do I do this? How can I tell when to use what first?
Best Answer
Try to only focus on one of them at a time. With differentiation, you usually work from the outside inwards, so figure out which part is the most inward so you can do that part last.
It's the power that is telling you that you need to use the chain rule, but that power is only attached to one set of brackets. It's the fact that there are two parts multiplied that tells you you need to use the product rule. Since the power is inside one of those two parts, it is going to be dealt with after the product.
So we'll do the product rule first.
Adding an extra set of brackets should help to know which part is "innermost": $$ f(x) = (x^3 +4)\left[ (4x^5 +2x-5)^\frac12\right] $$