oh! i wasnt trying to differentiate that equation
it started out as r=2+2cos(theta)
then i did y=rsin(theta)
dy/d(theta)=2cos(theta) + 2 cos^2(theta) - 2sin^2(theta) simplified down to
4cos^2(theta) + 2cos(theta) -2
so dy/dx = slope. i have to replace the thetas in 4cos^2(theta) + 2cos(theta) -2 with pi/2. but i dont know how to calculate 4cos^2(theta)......
Ok, so this approach is incorrect. i did the question a few posts up, refer to it to see the correct approach to this problem.
but anyway, to answer your question, you would calculate 4cos^2(theta) as 4(cos(theta))^2
so 4cos^2(theta)
= 4(cos(theta))^2
at theta = pi/2, we get
= 4(cos(pi/2))^2
= 4 (0)^2
= 0
so anyway, not that this is the right problem but dy/d(theta) = 4cos^2(theta) + 2cos(theta) -2 at theta = pi/2:
dy/d(theta) = 4cos^2(pi/2) + 2cos(pi/2) -2 = 0 + 0 - 2 = -2
which is not the answer you're looking for. i believe you said the answer was 1. you want to find dy/dx not dy/d(theta)
O ok, i get what you are trying to do now.
You want to find dy/d(theta) and dx/d(theta)
then dy/dx = dy/d(theta) * d(theta)/dx ..........since the d(theta)'s would cancel.
Ok, give me a sec and i'll post the solutions using your method
i take it you want a step by step walk-through when finding dy/d(theta) ?
now in polar coordinates:let t be theta
r = 2 + 2cost, find dy/dt when t = pi/2
y = rsint
x = rcost
let's find dx/dt
x = rcost
since r = 2 + 2cost
=> x = (2 + 2cost)cost
=> x = 2cost + 2cos^2(t) .........we have to use the chain rule on cos^2(t)
=> dx/dt = -2sint + 4cos(t)*(-sint)
=> dx/dt = -2sint - 4sintcost
now when t = pi/2
dx/dt = -2sin(pi/2) - 4sin(pi/2)cos(pi/2)
=> dx/dt = -2(1) - 4(1)(0)
=> dx/dt = -2 at t = pi/2
let's find dy/dt
y = rsint
since r = 2 + 2cost
=> y = (2 + 2cost)sint
=> y = 2sint + 2costsint
now we are going to find dy/dt. note that we have to find the derivative of 2costsint by using the product rule, since we have a product of two functions.
To refresh your memory, the product rule says:
if x and y are functions, then
(xy)' = x'y + xy'
that is we take the derivative of the first function times the second, plus the derivative of the second function times the first (or vice versa)
so for sintcost we have:
(sint)' = cost
(cost)' = -sint
so (sintcost)' = (sint)'cost + sint(cost)' = costcost + sint(-sint) = cos^2(t) - sin^2(t)
that was quite a side bar, let's go back to the problem
we left off at:
y = 2sint + 2costsint ..............now take the derivative
=> dy/dt = 2cost + 2(costcost - sintsint)
=> dy/dt = 2cost + 2cos^2(t) - 2sin^2(t) ..........now simplifying this any further is unnecessary, since it's not the function itself we are concerned about but it's numerical value, there's no need to waste time making this form look pretty (in case you are wondering though, your simplifications were correct)
now when t = pi/2
dy/dt = 2cos(pi/2) + 2(cos(pi/2))^2 - 2(sin(pi/2))^2
=> dy/dt = 0 + 0 - 2(1)^2
=> dy/dt = -2
now, derivative notations are more than notations, you can actually treat them like fractions. you can multiply them, cancel them out, seprate them, do whatever with them you can do with fractions. In that spirit, we will find dy/dx by the following.
dy/dx = (dy/dt)/(dx/dt) or you can say dy/dt = dy/dt * dt/dx ....same thing
dy/dt = -2
dx/dt = -2
so dy/dx = -2/-2 = 1 ............The answer
if you wanted to use dy/dt = dy/dt * dt/dx
dy/dt = -2
dt/dx = 1/-2
=> dy/dx = -2 * 1/-2 = 1
any questions? I'll have to answer them a bit later since i have class now
I have to adimit, the length of the methods are about the same, but this is easier than the method i originally proposed. my method you could do relatively quickly if you are good with implicit differentiation, but this way is much simpler--which is what math is all about, making life simple