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Use the Chain Rule to find $∂z/∂s$ and $∂z/∂t$.

$$ z = \tan(u/v) $$ $$ u = 5s + 7t $$ $$ v = 7s − 5t $$

This is the equation I used to find ∂z/∂s:

$$ \frac{\partial z}{\partial s}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial s}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial s} $$

Before substituting for $u$ and $v$, I had simplified to:

$$ \frac{\partial z}{\partial s}=\sec^2{\frac{u}{v}}\cdot(5-7\frac{u}{v^2}) $$

However, I can't see how I'll be able to get the equation in terms of $s$. Both $u$ and $v$ are functions on $s$ and $t$.

Michael Hardy
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Wilson
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  • Why are you not allowed to have $t$ in your answer? – David Sep 15 '14 at 23:30
  • I think you want $\frac{5}{v}$ instead of 5, and now you just want to substitute for u and v. – user84413 Sep 15 '14 at 23:44
  • I don't know enough to fully answer your question. However, there are several worked examples in my textbook that perform chain rule partial derivations that still have both variables within the answer – Adam Sep 16 '14 at 00:15

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why dz/ds should have function of s only?you have two independent variable and answer can be function of both of them.for better realize suppose we have a shape in x-y plane and we we know its velocity in cartesian coordinate now you want to obtain its velocity on polar coordinate we expect velocity for example in direction of radius have two Components.i hope you realized my sample. (sorry for my bad english)

Panda
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  • this is answer of question and my question isn't really question its Interrogative denial:D – Panda Sep 15 '14 at 23:55