Answer
$$\frac{\partial z}{\partial s}= t^2 \space cos(\theta) cos(\phi) - 2st \space sin(\theta)sin(\phi) $$
$$\frac{\partial z}{\partial t} = 2st \space cos(\theta) cos(\phi) - s
^2 \space sin(\theta)sin(\phi) $$
Work Step by Step
According to the Chain Rule:
$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial \theta}\frac{\partial \theta}{\partial s} + \frac{\partial z}{\partial \phi} \frac{\partial \phi}{\partial s}$$
$$\frac{\partial z}{\partial \theta} = (sin(\theta)cos(\phi))' = cos(\theta)cos(\phi)$$ $$\frac{\partial \theta}{\partial s} = (st^2)' = t^2$$ $$\frac{\partial z}{\partial \phi} = (sin(\theta)cos(\phi))' = -sin(\theta)sin(\phi)$$ $$\frac{\partial \phi}{\partial s} = (s^2t)' = 2st$$
$$\frac{\partial z}{\partial s} = (cos(\theta)cos(\phi))(t^2) + (-sin(\theta)sin(\phi))(2st)$$ $$= t^2 \space cos(\theta) cos(\phi) - 2st \space sin(\theta)sin(\phi) $$
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$$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial \theta}\frac{\partial \theta}{\partial t} + \frac{\partial z}{\partial \phi} \frac{\partial \phi}{\partial t}$$
$$\frac{\partial \theta}{\partial t} = (st^2)' = 2st$$ $$\frac{\partial \phi}{\partial t} = (s^2t)' = s^2$$
$$\frac{\partial z}{\partial t} = (cos(\theta)cos(\phi))(2st) + (-sin(\theta)sin(\phi))(s^2)$$ $$= 2st \space cos(\theta) cos(\phi) - s
^2 \space sin(\theta)sin(\phi) $$
