Use implicit differentiation to find y Use implicit differen

Use implicit differentiation to find y\'

Use implicit differentiation to find y\' square root 3xy=3x-6y x=y^4-4y^2

Solution

a) sqrt(3xy) = 3x -6y

d/dx(  sqrt(3xy)) = d/dx(3x -6y)

sqrt(3)* y/2sqrt(xy) + y\'(sqrt(3)*x/2sqrt(xy) = 3 -6y\'

y\' ( 6 + 3x/2sqrt(xy) = 3- sqrt(3)*y/2sqrt(xy)

y{ 12sqrt(xy) + 3x } = { 6sqrt(xy) -ysqrt3}

y\' = { 6sqrt(xy) -ysqrt3}/ { 12sqrt(xy) + 3x }

b) x = y^4 -4y^2

d/dx(x) = d/dx(y^4 -4y^2)

1 = 4y^3y\' -8yy\'

1 = y\'( 4y^3 -8y)

y\' = 1/( 4y^3 -8y)