How to differentiate? [Solved!]

X

Given: `x^3+x^2y+y^2 = 292`
Find: `dy/dx`, where : `x=600`, `y= 200`

Relevant page

<a href="/integration/integration-mini-lecture-find-y-given-dy-dx.php">7c. Given dy/dx, find y = f(x)</a>

What I've done so far

Watched the videos and read some of the pages. But I'm lost

X

Hi Oscar

There are examples similar to your question on this Implicit Differentiation page:

<a href="/differentiation/8-derivative-implicit-function.php">8. Differentiation of Implicit Functions</a>

Is that enough to get you started? 

Post your solution so we can see how you went. I encourage you to use the math entry system to make your math easier to read.

X

Yah - I can see it's like your examples.

`x^3+x^2y+y^2 = 292`

`3x^2 + x^2dy/dx + y(2x) + 2ydy/dx = 0`

`3x^2 + 2xy + (x^2  + 2y)dy/dx = 0`

`dy/dx = (3x^2 + 2xy)/(x^2  + 2y)`

Now substituting `x=600` and `y=200`

`dy/dx = (3(600)^2 + 2(600)(200))/((600)^2  + 2(200))`

`=3.66`

X

You almost got it right - your `dy/dx` is missing a negative...

X

Doh

Should have been:

`dy/dx = -(3x^2 + 2xy)/(x^2  + 2y)`

`dy/dx = -(3(600)^2 + 2(600)(200))/((600)^2  + 2(200))`

`=-3.66`

Thanks a lot.

X

For interest just now, I graphed your implicit function and noticed a problem.

<img src="/forum/uploads/imf-4548-x3x2yy2is292-graph.png" width="410" height="498" alt="How to differentiate?" />

The graph doesn't even go through the point `(600, 200)`!

This is a case where we can calculate an answer, but it doesn't have any meaning.

As you can see, the graph as `x` gets very big (and goes off in the negative direction) has slope very close to `-1`. 

When we substitute `x=200` into the original equation, we get `y^2+40000y+7999708 = 0` and this has solutions `y=-39799` or `y=-201.0`.

The slope is steeply negative for the first one, and very close to `-1` for the second.

Graphs tell us a lot about what is going on!

X

Thanks! I'll tell my teacher :-)