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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
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CHANGE OF VARIABLES Z X Y v u
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Differentiation of Implicit Function
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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
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Subtracting, we get
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Example 5: If z = ex sin y, where x = In t and y = t2, then find
Solution: We know that,
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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,
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Similarly Adding all the above, we get
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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
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Adding the result, we get
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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.
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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.
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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
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