Maths: Partial Differentiation

1) If x = rcos theta and y = r sin theta, show that partial r / partial x = cos theta and find partial theta / partial x.

2) If z = sin theta.sin phi.sin gamma, and z is calculated for the values theta = 30degrees, phi = 45 degrees and gamma = 60degrees, find approximately the change in the value of z if each of the angles theta and gamma is increased by the same small angle alpha degrees, and phi is decreased by 1/2 alpha degrees.

Can someone please show me how to work these out?

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  1. 1) change your equations to
    r^2 = x^2 + y^2
    2r dr/dx = 2x
    dr/dx = x/r = cos theta

    cos theta = x/r
    Now partially differentiate with respect to x.
    -sin theta d theta/dx = 1/r
    dtheta/dx = -1/(r sin theta) = -1/x

    I had to use d for the slanted Greek partial symbol above

    2) Assume alpha is very small and use
    dz = (partialz/dtheta)*alpha + (partialz/dphi)*alpha/2 + (partialz/dgamma)*alpha
    dz is the "total differential" increase in z.

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