Implicit Differentiation

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!

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