Lets say we have $f'(x)$ when $f(x)=(x^2+3)(x^3-1)$.
We could use product rule with $u = (x^2+3)$ and $v = (x^3-1)$, but we would get the same answer if we had just multiplied $uv$ before taking the derivative.
Does this apply to any problem where we take the derivative of two factors being multiplied and why? If so, why would we ever need to use the product rule? Does the same apply for the quotient rule (although i understand that for quotient rule, it would be easier to simplify when not dividing variables)?
Best Answer
Sure, you are always free to make any valid algebraic simplification at any time, e.g. expanding a product of polynomials. So a problem like this one can be done either by using the product rule, or by first multiplying out the polynomials and then using just the power and sum rules. Both methods will yield the same answer.
Well, think about an example like
$$\frac{d}{dx} \left[(8x^6+3x^5-2x^4+x^3-9x^2+2x+3)(2x^5+x^4-x^3+4x^2-8x+4)\right]$$
You can certainly multiply out all 42 terms if you like, but I think you would find it much more convenient to use the product rule here.
Also, not all functions are polynomials; if you want to find $\frac{d}{dx} x \sin(x)$ there is not really an obvious way to "multiply it out".
Sure, if the quotient can be simplified. But this is not always possible; if you look at $\frac{d}{dx} \frac{x^2+2x+2}{x^2+7x+12}$, the numerator and denominator have no common factors, so you can't really write it in a simpler way.