Question

Okay, so here's the thing. I'm REALLY REALLY CONFUSED BY IMPLICIT DIFFERENTIATION!!! The FUNCTION: \(\ \large x^3+x^2y+4x^2=6 \). Find \(\ \frac{dy}{dx} \). I've been on this problem for some 45 minutes, and have gotten no where on this homework assignment (this is the first problem). So, PLEASE PLEASE PLEASE HELP ME!!! SHOW me STEP-BY-STEP PLEASE!!!!!!!!!!!! Thank you so much in advance!!!!!

Answer 1

(I tried solving this problem, but I got a drastically incorrect problem! D: )

Answer 2

answer* not problem

Answer 3

What implicit diferentiation does, is the same thing as the normal diferentiation. You diferentiate on both sides in respect to one variable, and the derivative of the other variable that depends on the first appears isolated in normal diferentiation, and not isolated in implicit, you only need to isolate it.
In your function, you first diferentiate on both sides in respect to x, you can of course isolate y, wich is easy in this function, but lets do it in the way the problem wants to.
Diferentiating on both sides, what have you got?

Answer 4

Ummmm......
\(\ \Large 3x^2+2xy+8x=0? \)

Answer 5

@ivanmlerner

Answer 6

Ok, everything is correct but the second term. Now, remember that y is a function of x and therefore cannot be considered a constant.

Answer 7

Ok... So now what do I do

Answer 8

Try to diferentiate the second term considering that, do you remember what you do when you are diferentiating a multiplication of functions?

Answer 9

Product Rule? But that's where I get stuck with this problem. HOW do I do that?

Answer 10

Let's do just the \(x^2 y\) as an example.

First, we remember the product rule d(u v) = u dv + v du

second, we take the derivative with respect to x
\[ \frac{d}{dx} x^2y= x^2 \frac{d}{dx}y + y \frac{d}{dx}x^2\]
look carefully at this expression
\[ \frac{d}{dx}y \text{ is just } \frac{dy}{dx}\]
and
\[ \frac{d}{dx}x^2 \text { is } 2 x \frac{d}{dx}x= 2x \frac{dx}{dx} = 2x\]
we find
\[ \frac{d}{dx} x^2y= x^2 \frac{dy}{dx} + 2xy \]

Answer 11

Hmmm. So the derivative of \(\ \Large x^2y \text{ is } x^2\frac{dy}{dx}+2xy \text{ ?} \)

Answer 12

Wouldn't that be \(\ \Large 2x+2xy \text{ ?} \)

Answer 13

yes, I hope you are able to see how to get it.

Answer 14

I was able to follow along, thank you for that explanation @phi. Now, how do I proceed solving the original problem?

Answer 15

Finish the derivative (they are all just x terms so they are what you are used to...)
then "solve" for dy/dx

First what do you get for the derivate?

Answer 16

I haven't finished the problem, Ill try to solve it now..

Answer 17

I get \(\ \Large \frac{dy}{dx} = -\frac{3}{16x^2y}\)

Answer 18

How?!

Answer 19

remember
\[ \frac{d}{dx} x^2y= x^2 \frac{dy}{dx} + 2xy \]

Answer 20

But the correct answer is:
\(\ \Large y'=\frac{-x(3x+2y)}{x^2+8y} .... \text{ I don't know what I did wrong!!! :(}\)

Answer 21

Here's what I did: