show that dy/dx of equation 2x+y/x-5y=1 is equal to -1/6

Question

show that dy/dx of equation 2x+y/x-5y=1 is equal to -1/6

Answer 1

2x+y/x-5y=1

2x+y=x-5y

x=-6y now take the derivative with respect to x

1=-6dy/dx

dy/dx= -1/6

Answer 2

Implicit differentiation can help us solve this problem.

dy/dx (2x+y/x-5y = 1) multiply each side by (x-5y)

dy/dx ( 2x+y = x-5y)

d/dx(2x) + dy/dx (y) = d/dx (x) - dy/dx (5y)

2 + 1 dy/dx = 1 - 5 dy/dx move dy/dx one side

dy/dx + 5 dy/dx = -2 +1

6 dy/dx = -1

dy/dx = -1/6

Answer 3

(2x+y)/(x-5y) = 1

[(x-5y)(2 + y') - (2x+y)(1-5y')] / (x-5y)^2 = 0

denominator vanishes

2x - 10y + xy' - 5yy' - 2x + 10x y' -y + 5 yy' = 0

2x cancels, 5 yy' cancels

-10y + xy' + 10x y' - y = 0

-11y + 11 x y' = 0

x y' - y = 0

y' = y/x