Find the derivative
(a) $\displaystyle\frac{\partial w}{\partial s}$, where $\displaystyle w = \frac{x-z}{y+z}$, $x =s + t$, $y = st$ and $z = s - t$
Solution attempt:
$$\frac{\partial w}{\partial s} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial s} = \frac{1}{y+z} \cdot (1) + \frac{z-x}{(y+x)^2}t + \frac{-(x + y)}{(y + z)^{2}} (1) = (-x + z) \left(\frac{t}{(x + y)^2} + \frac{1}{(y + z)^2} \right)$$
(b) $\frac{\partial w}{\partial r}$, where $w = \sqrt{x^2 + y^2 +z^2}, $y = rs$, x = st, z = rt$
Solution attempt:
$$\frac{\partial w}{\partial r} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial r} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial r} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \cdot 0 + \frac{y}{\sqrt{x^2 + y^2 + z^2}} (1) + \frac{z}{\sqrt{x^2 + y^2 + z^2}}(1) = \frac{y+z}{\sqrt{x^2 + y^2 + z^2}}$$
(c) $\frac{\partial f}{\partial z}$, where $u = f(v)$, $v = g(w, x, y)$, $w = h(z)$, $x = p(t, z)$ and $y = q(t, z)$
Solution attempt:
$$\frac{\partial f}{\partial z} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial z} + \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial z} + \frac{\partial f}{\partial w} \cdot \frac{\partial w}{\partial z} + \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial z} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial z} = ???$$
Would this be right. Also, how would I get the derivative of the functions?
For more variables, more functions and more dependencies, the partial derivatives branch out into a tree, and are not evaluated in all variables at once as your attempt. Using subscript notation $$f_z=f_vv_z =f_v(v_ww_z+v_xx_z+v_yy_z)$$ and then using partial notation $$\frac{\partial f}{\partial z}=\frac{\partial f}{\partial v}\left(\frac{\partial v}{\partial w}\frac{\partial w}{\partial z}+ \frac{\partial v}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial v}{\partial y}\frac{\partial y}{\partial z} \right)$$ Your answers for the first two parts are correct.