If you are given the function f x y w xy w3 8xyw 3w with

If you are given the function f (x, y, w) = xy / w3 - 8xyw + 3w with x = r-s-t, y = 4r + 2s + 3t. W = r2 + s2 + t2, then find . Note: you do NOT have to simplify your answer, and your final answer may contain the variables x, y, was well as r, s, t.

Solution

for a function of two or more variables we can write df=(df/dx)dx+(df/dy)dy+(df/dw)dw now to find df/ds we use chain rule for a function g of one variable we write dg/ds=(dg/dx)*(dx/ds) by chain rule similarly here when we use chain rule we get df/ds=(df/dx)dx/ds+(df/dy)dy/ds+(df/dw)dw/ds hence df/ds=-(y/w^3-8yw)+2(x/w^3-8xw)+2s(-3xy/w^4-8xy+3)