Question

How do you implicitly differentiate `sqrt(3x+3y) + sqrt(3xy) = 17.5`?

Answer 1

`sqrt(3x+3y)+sqrt(3xy) = 17.5`

`sqrt(3x+3y)+sqrt(3xy) = sqrt3 sqrt(x+y)+sqrt3 sqrt(xy)`, so

`sqrt(3x+3y)+sqrt(3xy) = 17.5` is equivalent to

`sqrt(x+y)+sqrt(xy) = 17.5/sqrt3`

Differentiate both sides with respect to `x`, using the chain rule.

`1/(2sqrt(x+y))(1+dy/dx) + 1/(2sqrt(xy))(y+x dy/dx) = 0`

`1/(2sqrt(x+y))+ 1/(2sqrt(x+y))dy/dx + y/(2sqrt(xy))+x/(2sqrt(xy)) dy/dx) = 0`

`((sqrt(xy)+xsqrt(x+y))/(sqrt(xy)sqrt(x+y)))dy/dx = (-sqrt(xy)-ysqrt(x+y))/(sqrt(xy)sqrt(x+y)`

`dy/dx = (-sqrt(xy)-ysqrt(x+y))/(sqrt(xy)+xsqrt(x+y))`