differential equation with substitution

dy/dz isnt right as
y = z^-1/2
dy/dz = -1/2 z^-3/2

Looks like you put in dz/dy rather than dy/dz.

what? where is dy/dz?

Im presuming you did the chain rule for dy/dx
dy/dx = dy/dz dz/dx = ...
You should be clear about what you tried to do there anyways.

oh yeah with chain rule it makes sense but do i have to do chain rule in these kind of questions? i thought chain rule was not needed, i always differentiated y with respect to x and then just put the d?/d? something of the equation i need to get to in this case i have to put dz/dx next to the derevative is that wrong?

Not sure I fully understand what youre saying, but yes you need to explicitly use the chain rule as youre introducing and transforming the equation to another variable z so you think like y(z(x)) so
dy/dx = dy/dz dz/dx
and you end up with dy/dx replaced by dz/dx (and similar for higher derivatives) multiplied by an expression which in this case cancels with the y^3 on the right.

You wrote
dy/dx = dz/dy dz/dx
which isnt correct

in this question how do i usse chain rule?
for part a

Same again. dy/dx = (dy/dz)(dz/dx)

when do i not do chain rule then, i think there is a case where it is not needed

In the context of this thread, you use the chain rule as above. When do you think otherwise?

i asked my teacher and she told me i need chain rule only when i dont have the verible i need for the question