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CHANGE OF VARIABLES

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. v. u. Differentiation of Implicit Function.

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CHANGE OF VARIABLES

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  1. DIFFERENTIATION OF COMPOSITE FUNCTIONLet 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

  4. 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

  5. Subtracting, we get

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

  7. 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,

  8. Similarly Adding all the above, we get

  9. 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

  10. Adding the result, we get

  11. 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.

  12. 6. If 7. If 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.

  13. 12. If z=eax+byf(ax-by), prove that , show that 13. If 14. Find dy/dx if (i) x4+y4=5a2axy. (ii) xy+yx=(x+y)x+y

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