Differentiation

(1)
y=x(2x-3)^(1/2)
Let a=(2x-3)^(1/2)
da/dx=da/d(2x-3)*d(2x-3)/dx
=1/2(2x-3)^(1/2) * 2
=(2x-3)^(-1/2)
dy/dx=a(dx/dx)+x(da/dx)
=(2x-3)^(1/2)+x(2x-3)^(-1/2)

(2)
y=[(2x-3)^4][(3x-2)^2]/3
Let a=(2x-3)^4,b=(3x-2)^2
da/dx=da/d(2x-3)*d(2x-3)/dx
=4(2x-3)^3*2
=8(2x-3)^3
db/dx=db/d(3x-2)*d(3x-2)/dx
=2(3x-2)*3
=6(3x-2)
dy/dx=[a(db/dx)+b(da/dx)]/3
={[(2x-3)^4][6(3x-2)]+[(3x-2)^2][8(2x-3)^3]}/3
=[6(3x-2)(2x-3)^4+8(3x-2)^2 (2x-3)^3]/3
=(3x-2)(2x-3)[6(2x-3)^3+(3x-2)(2x-3)^2]/3

1. y = x(2x-3)^1/2

dy/dx = (2x-3)^1/2 + x(1/2)(2)(2x-3)^-1/2

= [(2x-3)^1/2][(2x-3)+x]

= [(2x-3)^1/2](3x-3)

= 3[(2x-3)^1/2](x-1)



2. y = [(2x-3)^4][(3x-2)^2/3]

dy/dx = [4(2)(2x-3)^3](3x-2)^2/3 + [(2x-3)^4](2/3)[(3x-2)^-1/3](3)

= 2[(2x-3)^3][(3x-2)^-1/3][4(3x-2)+(2x-3)]

= 2[(2x-3)^3][(3x-2)^-1/3][14x-11]

1, X(2X - 3)^1/2

its derivative = (2X - 3)^1/2+ X * (1/2) * 2*(2X - 3)^(-1/2) = (2X - 3)^1/2+X(2X - 3)^(-1/2)


2, (2X - 3)^4(3X - 2)^2/3

its derivative = 4*3*(2X - 3)^3(3X - 2)^2/3 + 3*(2/3)*(2X - 3)^4(3X - 2)^(-1/3) = 12 (2X - 3)^4(3X - 2)^2/3 + 2 (2X - 3)^4(3X - 2)^(-1/3)