# Thread: quick implicit differentiation question

1. ## quick implicit differentiation question

ok just a question about the working out for this problem,

I understand

4x^2 - 8xy - 9y^2 = -637

= 8x - 8y - 8x d/dx(y) - d/dy(9y^2)dy/dx

= 8x - 8y - 8x dy/dx - 18y dy/dx

but where does the + sign come from??? can anyone explicitly show where this comes from.. I think it must come from the product rule but I just cant see it...

(8x-8y)-(8x+18y)dy/dx

2. They just factored it.

3. Originally Posted by dankelly07
ok just a question about the working out for this problem,

I understand

4x^2 - 8xy - 9y^2 = -637

=> 8x - 8y - 8x d/dx(y) - d/dy(9y^2)dy/dx = 0

=> 8x - 8y - 8x dy/dx - 18y dy/dx = 0

but where does the + sign come from??? can anyone explicitly show where this comes from.. I think it must come from the product rule but I just cant see it...

=> (8x-8y)-(8x+18y)dy/dx = 0
NB: I realise what you're trying to say. Nevertheless, the editing (in red) I've done is essential - otherwise what's posted makes no mathematical sense. I can assure you that if this was an answer submitted in an examination or assignment it would lose marks if given in its original form.

4. ok now I'm wondering for a implicit differentiation problem, how i determine a local max, local min or if the case is degenerate? any help would be ace..

5. A local minimum is a point c such that $\displaystyle f(c) \le f(x)$ for all x in an open interval containing c.
A local maximum is a point c such that $\displaystyle f(c) \ge f(x)$ for all x in an open interval containing c.

Best points to focus on is the critical points, the points where the derivative of the function does not exist or equals 0. Plug these critical points in the original function.

6. $\displaystyle \frac{{dy}} {{dx}} = \frac{{8x - 8y}} {{8x + 18y}} = 0$

$\displaystyle \begin{gathered} 8x - 8y = 0 \hfill \\ = > x = y \hfill \\ \end{gathered}$

$\displaystyle \begin{gathered} 4x^2 - 8x^2 - 9x^2 = - 637 \hfill \\ - 13x^2 = - 637 = > x^2 = 49 \hfill \\ x = \pm 7 \hfill \\ ( - 7, - 7)and(7,7) \hfill \\ \end{gathered}$

so are these the 'critical points'? is the first point a local min and the second a local max?