Question

Find the instantaneous rate of change of w with respect to z if w=7/3z^2.

Answer 1

They want us to find \[\large \frac{dw}{dz}\]
So we need to take the derivative with respect to z. Confused about how to do that? :o Have you learned about derivatives yet?

Answer 2

Well I know how to take derivatives, I just wasn't sure about how to go about it. Would I use the... well, I think it's called the quantitative rule but I'm not sure. It's ba'-ab'/b^2. So that would be ((3z^2*0)-(7)(6zz'))/3z^4, right?

Answer 3

Without simplifying, of course.

Answer 4

The Quotient rule? You certainly could, but that seems unnecessary since the top term is a constant.

It's probably a little bit better of an idea to - rewrite the z term with a negative exponent. Remember how to do that? You bring it up to the top from the denominator, and change the sign on the exponent.

Answer 5

\[\huge \frac{7}{3z^2}\qquad \rightarrow \qquad \frac{7z^{-2}}{3}\]Which can written even nicer like this,\[\huge \frac{7}{3}z^{-2}\]From here it's just the power rule.

Answer 6

The way you did it looks good though :) I don't see any mistakes.
So if that method makes more sense to you, then by all means stick with it.

Oh just remember that when you SQUARE the b term in the bottom, you square the entire b, which would include the 3.

Answer 7

So maybe one lil mistake heh.

Answer 8

Ahh okay, I didn't even think of it that way! I forget about all of my options sometimes, haha. So, simplified it would be\[w'=\frac{ -14z' }{ 3z^3 }\]
Right?

Answer 9

Yah looks good c:

Answer 10

Okay, thanks so much! :)