The Product Rule

A product of two functions changes for two separate reasons at once: f is changing, and g is changing. If you only tracked one of them you'd miss half the story, so the derivative has to account for both contributions and add them together. That's the whole idea behind f'g + fg'.

Picture a rectangle with width f(x) and height g(x), so its area is f(x)g(x). Nudge x forward a little. The width grows by some small amount and the height grows by some small amount, so the area gains a thin strip along the top (width times the change in height) and a thin strip along the side (height times the change in width). The tiny corner piece where both change is negligible for small nudges. Divide those two strips by the nudge in x and you get exactly f'(x)g(x) + f(x)g'(x) - the rectangle picture is the product rule.

This rule is the reason you can't just differentiate factor by factor. It sits next to the quotient rule, which is really the product rule applied to f times g⁻¹, and it combines constantly with the chain rule once your factors are themselves compositions, like x³ sin(2x). Later, when you build up to products of three or more functions, or to implicit differentiation, you'll keep reaching for this same pattern: hold one piece still, differentiate the other, then swap.