I dont understand how to do this partial derivative

Question

If z=f(x,y), where x=rcosϴ and y=rsinϴ, find ∂²z/ ∂r∂ϴ. 

 

I have the complete solution in my book but have cant understand anything.Please help !

Answer 1

If the function z(x,y) is "well-behaved" then

 

∂²z/∂r∂ϴ = ∂/∂θ (∂z/∂r) = ∂/∂r (∂z/∂θ),

 

i.e., the order of differentiation is interchangeable.  Just differentiate z with respect to r, and then differentiate the result with respect to θ, or do this in opposite order.  But first you must express z as a function of r and θ.  You will need to replace the x in z with rcosθ and the y in z with rsinθ.  If you give us the actual function z(x,y) we could help you more.