Use the power rule to differentiate the following functions below.
(1) f(x) = sqrt{9 - 4x}
(2) f(x) = 7x^3 - 3
I'll just help ya for the first problem because the second is very easy.
Put $\displaystyle g(x)=9-4x$ so that $\displaystyle f(x)=\sqrt{g(x)}=g(x)^{1/2}$ and now we can apply the power rule but, always! having in mind the chain rule, thus $\displaystyle f'(x)=\frac12g(x)^{-1/2}\cdot g'(x).$ Finish it!
I am by no means as intelligent as yourself but why would you even give an answer in that form? I can apply the power rule without the chain rule and I do know how to use the chain rule but why wouldn't you just say:
Power rule: $\displaystyle n(f(x)^{n-1})*f'(x)$ which gives us the correct answer of $\displaystyle -4*\frac{-1}{2}(9-4x)^{-1/2}$ which simplifies to $\displaystyle \frac{-2}{\sqrt{9-4x}}$. Your answer is visually far more complex and another thing, not that you really used any, but it seems like the vast majority of people do not know math notation. I think with some of the answers a KISS approach would really help some people. Sorry for posting in an ancient thread that someone revived.
\rant
Neat, but it still requires the chain rule.
It was just a notational convenience which might be easier to understand for someone who has not learned the power rule in its most general form. Krizalid was trying to more clearly show the chain rule's involvement.
I realize that, however, my point really is that if the question is something you would normally do in your head the person posing question would probably like the simplest answer/explanation possible and it seems like people often get the complete opposite. I also think some of these people are too damn smart!
What do you propose? The responder simply say, "Okay, clearly this can be done in your head and you will get $\displaystyle \frac{-2}{\sqrt{9-4x}}$"?
Yes, it can be much easier to do these sorts of tasks without writing them out, but the extra exposition can be helpful for someone who is not as experienced. As the individual gains in experience, he or she will gradually need to write out less steps.