This is a very quick question, and it might be pretty basic. But as a preface, I plan to dive into the actual proofs behind the derivative rules after this post, but I would like to see if my intuition is correct here first!
I was toying around with derivative rules, and I had the thought that, no matter what form your function is in, you will be performing the product rule!
For example, if you want to take a derivative of the function $f(x) = x^2\times sin(x)$, you would apply the product rule and get $\frac{d}{dx} f(x) = 2x\times sin(x) + x^2\times cos(x)$
This is an obvious use of the product rule, but I think that there is a less obvious use of the product rule when taking the derivative of $f(x) = x^2$. The derivative is simple, if we use the power rule we get: $$\frac{d}{dx} f(x) = 2x$$
And this is done with the formula $\frac{d}{dx} [x^n] = nx^{n-1}$.
But I am positing that there is a "hidden" product rule being performed as well! Because if we think about it, there is still multiplication taking place with the $x$ in $f(x) = x^2$, this is shown in the "expanded" version of: $$f(x) = 1\times x^2$$
And you cannot just skip over the product rule! So in taking the derivative of this function we are actually performing a combination of the power rule and product rule: $$\frac{d}{dx} f(x) = 0\times x^2 + 1\times 2x$$
which is implicitly done with the output of the power rule alone, so this is why we do not show the product rule in action here.
This is my own intuition based on what I have learned so far, and I am curious if I am correct, or making dangerous assumptions!
Best Answer
Great question. Your assumptions are correct. Your work isn't too important for derivatives, but when you get to integral calculus keep your intuition in mind and you will be ahead of your class. I will put how in a spoiler below, but I highly suggest you don't click on it unless you're burning to find out. Also, your intuition is probably the way the person who first made the product rule tested whether it worked or not.