Parametric equation/chain rule question?

Question

We are supposed to find slope of the parametric equations using implicit differentiation with the given value of T.

We are given: X = sqrt((5-sqrt(T))) , Y(T-1) = sqrt(T) and T=4

Note that the X portion of the equation reads as X equals 5-T all under the square root, and T by itself under an additional square root within the original square root.

So far, what I have for dx/dt is, using the chain rule:

dx/dt = 1/2((5-T^(1/2)))^(-1/2) * ((-1/2T^(-1/2)))

After multiplying I got,

dx/dt = -1/4T((5-T^(1/2)))^(-1)

by multiplying the -1/2T by the 1/2, and adding the exponents of (-1/2) + (-1/2) = -1

Is this the right way to multiply the terms?

After working through the entire problem, evaluating dy/dt divided by dx/dt using the values for X, Y, and T, and assuming that is the correct way to find dx/dt, I got 5/12 as the slope.

Mathsman: did you mean to type out Y+ dy/dt(T-1) = 0.5T^(-0.5) because the Y side of the equation is given as Y(T-1)
=sqrt(T), not (T+1).

Which would result in Y = 2/3 with T = 4

Because 4-1 is 3, so I have 3Y on the left side, and 2 on the right side from the sqrt of 4 = 2. So wouldn't Y be 2/3 and not 2/5?

Answer 1

I normally don't give specific answers but help with how to get them, but this one must be an exception as you will see.

The previous answer starts correctly and gets the right value for dx/dt although in a long-winded way. However, it contains an error in the middle.
y(t + 1) = sqrt(t) does not lead to the next line given. (It would lead to y = sqrt(t)/(t + 1) if necessary but it isn't needed.)
y(t + 1) = sqrt(t) done by implicit differentiation gives
y + dy/dt(t + 1) = 0.5t^(-0.5)

So t = 4 gives 5y = 2 ---> y = 2/5 and
2/5 + 5dy/dt = 1/4 ---> dy/dt = -3/100

It is also far easier to do the x differentiation implicitly.
x = sqrt(5 - sqrt(t)) ---> x^2 = 5 - sqrt(t)
---> 2x*dx/dt = -(0.5)*t^(-0.5)
When t = 4 ---> x = sqrt(5 - 2) = sqrt(3)
---> 2sqrt(3)*dx/dt = -(0.5)/2 ---> dx/dt = -sqrt3/24

So we finish up with dy/dx = (dy/dt)/(dx/dt)
= (-3/100)/(-sqrt3/24) = 6sqrt(3)/25
Note that it is fairly easy to get y as an explicit function of x to check this result.

Answer 2

Given:

x(t) = √(5 - √t)
y(t - 1) = √t
t = 4

Find dy/dx at t = 4

We have

dx/dt = {1/[2√(5 - √t)]} [-1/(2√t)] = -1 / [4(√t)√(5 - √t)]
dx/dt = -1 / {4√[t(5 - √t)]}
dx/dt = -1 / {4√[4(5 - √4)]} = -1/(4√12) = -1/(8√3)

y(t - 1) = √t
y(t) = √(t + 1)

dy/dt = 1 / [2√(t + 1)]
dy/dt = 1/[2√(4 + 1)] = 1/(2√5)

dy/dx = (dy/dt) / (dx/dt) = [-1/(8√3)] / [1/(2√5)]
dy/dx = -2√5 / (8√3) = -√5 / (4√3)
dy/dx = -√15 / 12