Using chain rule to find formula

Question

Let f(x,y,z) = e^x+ysinz, and let x= g(s,t), y=h(s,t), z=k(s,t) and m(s,t)=f(g(s,t),h(s,t),k(s,t)). Find a formula for m_st using the chain rule and verify that your answer is symmetric in s and t.

Answer 1

The best way to see this is to use a tree diagram, but it is hard to draw one in a chat box, so I try to explain without one.

 

For this problem, you have f as a function of x, y, and z; and x, y, z as functions of s and t.

 

m_st is the second partial derivative of m with respect to s first, then t second.

 

You must first find m_s using the Chain Rule:

 

m_s = f_x x_s + f_y y_s + f_z z_s

 

You must write the answer completely in terms of s and t only, before proceeding.

 

After this, you can then find the partial derivative of THIS function w.r.t t, and then simplify as usual.

 

I suggest you try to finish this without any more suggestions from me or anyone else, then reply back if you need more help. It is better to finish the problem with a little guidance rather than just have someone give you the answer.

 

I will get you reply if you need more help. Good luck!