If $u=\arctan(xy+z)$, where $$x=s^2+t^2,\;y=9re^{st},\;z=r^2st,$$ find the value of $\frac{\partial u}{\partial s}$ when $r=2,s=1,t=0$.
Is my attempt so far correct?
$$\frac { ∂u }{ ∂s } =\frac { ∂\tan^{ -1 }(xy+z) }{ ∂s } \\[12pt] =\frac { ∂\tan^{ -1 }(xy+z) }{ ∂(xy+z) } \frac { ∂(xy+z) }{ ∂s } \\[12pt] =\frac { 1 }{ 1+{ (xy+z) }^{ 2 } } \frac { ∂(xy+z) }{ ∂s } $$
Simplifying the factor on the right:
$$\frac { ∂(xy+z) }{ ∂s } \\[12pt] =\frac { ∂(xy+z) }{ ∂x } \frac { ∂x }{ ∂s } +\frac { ∂(xy+z) }{ ∂y } \frac { ∂y }{ ∂s } +\frac { ∂(xy+z) }{ ∂z } \frac { ∂z }{ ∂s } $$
Continuation of attempt: http://s16.postimg.org/4lk9voaxx/IMG_20130415_173718.jpg Sorry for poor quality.
