Q: Using implicit differentiation, find dy/dx of
y/(x+6y)=x7-9
at the point (1,-8/49)
Answer:
Taking the derivative of the right side of the equation is fairly basic, taking the derivative of the left side, on the other hand, is harder. The left side involves a quotient rule. I am going to ignore the right side of the equation for now and just deal with the left:
y/(x+6y)
The quotient rule tells you (in some form / notation or another):
(bottom)*(deriv. of top) – (deriv of bottom)*(top) all divided by (bottom) squared. [You may have learned this with f’s and g’s and f ‘ and g ‘ — it all says the same thing]
So:
Top = y
Bottom = x + 6y
Derivative of Top = dy/dx = y’ [whatever notation you are using]
Derivative of Bottom = 1 + 6(dy/dx) = 1 + 6y’
So, plug this into the quotient rule to get:
[(x+6y)*y’ – (1+6y’)*y] / (x + 6y)2
That is the derivative of the left side of the equation. Take the derivative of the whole thing (left and right) and you get:
[(x+6y)*y’ – (1+6y’)*y] / (x + 6y)2 = 7x6
We now have our derivative. We need to solve for y’ by plugging in the point that was given to us: (1,-8/49)
Yuck. This is a lot of algebra. I don’t have time right now to show all the algebra, but when I plug in the point I get the following:
y’ = -55/343
I checked this solution by hand and on a computer, so I am fairly confident — double check it yourself!
