The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$
The answer is apparently
$$(z/z^2)(2\sin{t}\cos{t}) - (z/z^2)(2\sin{t}\cos{t}) + ((-x-y)/z^2)(-2/4t^2)$$
i get everyting except where the $$(z/z^2)$$ comes from. should the partial derivative of x and y just be (1/z)?
I'm guessing it was $f(x,y,z) = x/z + y/z$
You should have $\frac {\partial f}{\partial x}=\frac 1 z=\frac z {z^2}$
Be careful in your notation later on. You have written $...+ (-x-y/z^2)(-2/4t^2)$
but it should be $...+ ((-x-y)/z^2)(-2/4t^2)$
or better still: $...+ \left (\frac {-x-y} {z^2} \right) \left( \frac {-2} {4t^2}\right)$