Use the chain rule to find (∂ z)/(∂ s) and (∂ z)/(∂ t), z =...

Use the chain rule to find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$, $\qquad z = \frac{\sin(\theta)}{r}$, $r = st$, $\theta = s^5 + t^5$ $\frac{\partial z}{\partial s} = \text{________} $\frac{\partial z}{\partial t} = \frac{9t \cos(s^9 + t^9)}{s} - \frac{\sin(s^9 + t^9)}{st^8}$

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00:01 Hi, in this question we are given with z to be equals to e raised to par r, cost of theta, r is given to be equals to s of t and theta to be equals to under root of s raise to par 8 plus t t raise to par 8.

00:17 Now from here, as we know, we need to find curly z by curly t and curly z by curly s.

00:24 For that, we will find the curly r by curly s that will be equals to t and curly r by curly t that will be equals to s.

00:37 Also, curly theta by curly s will be equal to derivative of this with respect to s, which is 1 by 2 under root of s raise to par 8 plus t raise 2 times derivative of this which will be 8 times s raise to par 7, where this 2 will cancel.

00:55 This 8 by 4 and we get 4 s to as to per 7 by under root of s 2 to par 8 plus t raise 2 par 8 similarly if we differentiate with respect to t we get this to be equals to 4 times t r 7 by under root of s 2 2 2 2 2 2 2 2 2 4 8 now we will solve for curly z by curly t that will be equals to derivative with respect to t of e r cost theta here we'll apply chain rule and this will be equals to first derivative of second which is minus sine of theta times curly theta by curly t plus cos theta derivative of e raised 2 par r which is e raised 2 per r d r d t now substituting the known values here we get this to be equals to minus e raised to par r sine of theta can be replaced by under root of t raised 2 % plus s r.

01:53 R.

01:54 Kurly theta by curly t that we have calculated as four times t.

01:59 R t.

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01:59 7 by under root of s.

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02:02 S.

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02:03 Kroat of s.

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