1. DIFFERENTIATION OF COMPOSITE FUNCTION Let z = f ( x, y) Possesses continuous partial derivatives and let x = g (t) Y = h(t) Possess continuous derivatives z y x t

2. CHANGE OF VARIABLES Z X Y v u

3. Differentiation of Implicit Function

9. Example 4: z is a function of x and y, prove that if x = eu + e-v, y = e-u + e-v then
Solution: z is a change of variable case

10. Subtracting, we get

11. Example 5: If z = ex sin y, where x = In t and y = t2, then find
Solution: We know that,

12. Example 6: If H = f(y-z, z-x, x-y), prove that
Solution: Let, u = y-z, v = z-x, w = x-y
→ H = f(u,v,w)
H is a composite function of x,y,z. We have,

13. Similarly
Adding all the above, we get

14. Example 7: If x = r cosθ, y = r sinθ and V=f(x,y),
then show that
Solution: We have, x = r cosθ, y = r sinθ

18. Adding the result, we get

23. Exercise 1. If z = xm yn, then prove that
2. If u = x2-y2, x=2r-3s+4, y=-r+8s-5, find
3. If x=r cosθ, y=r sinθ, then show that
(i) dx = cos θ.dr - r sin θ.dθ
(ii) dy = sin θ.dr + r.cos θ.dθ
Deduce that
dx2 + dy2 = dr2 + r2dθ2
x dy – y dx = r2.dθ
4. If z = (cosy)/x and x = u2-v, y = eV, find
5. If z=x2+y and y=z2+x, find differential co-efficients
of the first order when
y is the independent variable.
z is the independent variable.

24. 8. If u = (x+y)/(1-xy), x=tan(2r-s2), y=cot(r2s) then find
9. If z=x2-y2, where x=etcost, y=etsint, find dz/dt.
10. If z=xyf(x,y) and z is constant, show that
11. Find and if z = u2+v2+w2, where
u=yex, y=xe-y, w=y/x.

25. 12. If z=eax+byf(ax-by), prove that
, show that If
Find dy/dx if
(i) x4+y4=5a2axy.
(ii) xy+yx=(x+y)x+y