I came across a homework problem in my calculus 1 class where I was supposed to find the derivative of a function $y=(4x+15)^3$. This in and of itself was pretty easy, however, I thought I noticed something odd when I worked through it. The answer to the problem is $\frac{dy}{dx}=12(4x+15)^2$, but this isn't equivalent to what you would have gotten if you had used $u$ substitution and applied the power rule such that: $$y=(4x+15)^3$$ $$u=4x+15$$$$y=u^3$$$$y=3u^2$$ $$y=3(4x+15)^2$$
Obviously this is incorrect, but conceptually, why doesn't this work? In my mind, they should produce the same answer. I did read a proof of the chain rule, which I understood just fine, but that didn't really answer the question.