Find the value of (dr)/dt at t=0 if r=(theta^2 +7)^(1/3) and theta^2t + theta =1?

So far, I have

(dr)/(d theta) = 1/3(theta^2+7)^(-2/3) * 2theta

and

(d theta)/dt = (2thetat + 1) / -theta^2

I tried to multiply them together, but that doesn't get me anywhere.

1 Answer
Jan 17, 2018

-1/6

Explanation:

You're definitely on the right track!

As you pointed out, we start with chain rule:
(dr)/(dt) = (dr)/(d theta) * (d theta)/(dt)

You calculated these values correctly.

At t=0, the second equation gives theta = 1. The second equation then gives that r = 2 at that time. So we can actually plug those values into the equations you supplied, yielding
(dr)/(d theta) = 1/3 * (8)^(-2/3) * 2 * 1 = 2/3 * 1/4 = 1/6

(d theta)/dt = (2 * 1 * 0 + 1)/(-1^2) = -1

Hence,
(dr)/(dt) = (dr)/(d theta) * (d theta)/(dt) = -1/6.