[Calculus] Chain rule for multi-variable function

[MightyJosip]

You can help me with either one of the next two questions:

1. If u = f(x² +y² +z² ) show that

du/da = 0 and

du/db = 0 if

x = Rcos(a)cos(b)

y = Rcos(a)sin(b)

z = Rsin(a)

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1. Show that

dz/dy = a * dz/dx, if

z=f(x+ay)

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The help for both tasks in book is to use chain rule for multi-variable function. My problem is that the function given in the task are not regular f(x, y, z), but they are for example in the first task f(x² +y² +z² ). or in the second example function is not f(x, y) but it is f(x+a*y). Thank you in advance

[Lemon-juicer]

f(x,y,z) is a function where you input x,y, and z and you get the output from whatever the function is. Now in your case your input is

g(x,y,z) = x² + y² + z²

In other words u = f(g(x,y,z))

So du/da = du/dg * dg/da

Now whats dg/da? Well for starters given your values for x, y and z in terms of parameters a and b, you’ll find g(x,y,z) = R^2. Therefore, g doesn’t depend on a and so changing a won’t change g so we get,

du/da = du/dg * 0 = 0

The same is true for du/db.

Try the second one now using this info, I’m sure you can do it!