DIFFERENTIATION OF COMPOSITE FUNCTION Let z f x

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

CHANGE OF VARIABLES Z X Y u v

Differentiation of Implicit Function

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

Subtracting, we get

Example 5: If z = ex sin y, where x = In t and y = t 2, then find Solution: We know that,

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,

Similarly Adding all the above, we get

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

Adding the result, we get

Exercise 1. If z = xm yn, then prove that 2. If u = x 2 -y 2, x=2 r-3 s+4, y=-r+8 s-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 (i) dx 2 + dy 2 = dr 2 + r 2 dθ 2 (ii) x dy – y dx = r 2. dθ 4. If z = (cosy)/x and x = u 2 -v, y = e. V, find 5. If z=x 2+y and y=z 2+x, find differential co-efficients of the first order when (i) y is the independent variable. (ii) z is the independent variable.

6. If 7. If 8. If u = (x+y)/(1 -xy), x=tan(2 r-s 2), y=cot(r 2 s) then find 9. If z=x 2 -y 2, where x=etcost, y=etsint, find dz/dt. 10. If z=xyf(x, y) and z is constant, show that 11. Find and u=yex, y=xe-y, w=y/x. if z = u 2+v 2+w 2, where

12. If z=eax+byf(ax-by), prove that 13. If 14. Find dy/dx if (i) x 4+y 4=5 a 2 axy. (ii) xy+yx=(x+y)x+y , show that