Differentiation of composite functions

If u=f(x_1 y)then total differential equation of ‘u’ is given by

Type 1 : Total derivative rule

If u=f(x,y) where x=x(t) and y=y(t),then

Type 2 : Total derivative Chain rule

If u=f(x,y) where x=x(r,s) and y=y(r,s),then
u is a function of (x,y),x and y are dependant on (r,s),hence
u becomes a function depndant on (r,s)

Total Derivatives Problems

  1. Find du/dt when u=x^3 y^2+x^2 y^3 with x=at^2, y=2at. Use partial derivatives.
  2. Find du/dt if u=xy+yz+zx and x=t cost , y=tsint , z=t @ t=π/4
  3. If u=f(x/y, y/z, z/x) prove that p=x/y, q=y/z, r=z/x
  4. If u=f(x-y, y-z, z-x), Show that del u/del x+del u/del y+del u/del z=0
  5. If z=f(x, y) where x=r cos(theta) and y=r sin(theta), Show that (del z/del x)^2 + (del z/del y) ^2 = (del z/del r)^2+1/r^2 . (del z/del (theta))^2
  6. If z = sin (ax+y) + cos (ax-y), Prove that (del)^2 z / del x^2 = a^2 (del)^2 z / del y^2
  7. If z(x+y) =x^2+y^2, Show that (del z/del x – del z/del y) ^2= 4(1- (del z/del x)-(del z/ del y))


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