Differentiating s = √(3x² + 6y²) with respect to time

[rectifryer]

Homework Statement

s=$\sqrt{(3x^2)+(6y^2)}$

Homework Equations

None

The Attempt at a Solution

$\stackrel{ds}{dt}$=$\stackrel{d}{dt}$$\sqrt{(3x^2)+(6y^2)}$

$\stackrel{3x}{\sqrt{(3x^2)+(6y^2)}}$

The problem with that is its only d/dx if y is a set number. I don't know how to differentiate the entire thing properly. I have been hacking at this for 8 hours. I feel like mental jello.

 

[Villyer]

You are taking the derivative with respect to t.

So d/dt of 3x² = 6x * dx/dt, not 6x.Maybe this helps figure out the whole derivative?

 

[Astronuc]

Well the relevant equation under 2. Homework Equations would be an expression of the chain rule.

d/dt(f(g(t)) = f'(g(t))*g'(t)

http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html
http://mathworld.wolfram.com/ChainRule.html

Let g(t) = g(x(t),y(t)) and f = √

One could also write the original equations as s² = 3x² + 6y², and differentiate each term with respect to t.

 

[rectifryer]

One could also write the original equations as s² = 3x² + 6y², and differentiate each term with respect to t.

That doesn't really seem like it would get me anywhere. I know I am wrong, but why would that work?

Thank you for the links.

 

[rectifryer]

I have narrowed down my question, specifically to the area I have highlighted on this picture (bear in mind, I can't post pics under 10 posts):

http:// i.imgur .com /62erw.png

Where did all the dx/dt and dy/dt come from on the right side? I don't understand that step. I know how to do this when thinking about it in function form, but it confuses me to think about it in fraction form, which is what's required to answer.

 

[HallsofIvy]

If s is a function of two variables, x and y, which are themselves functions of t. The "chain rule" says
$$\frac{ds}{dt}= \frac{\partial s}{\partial x}\frac{dx}{dt}+ \frac{\partial s}{\partial y}\frac{dy}{dt}$$

Here, $s(x,y)= \sqrt{3x^2+ 6y^2}= (3x^2+ 6y^2)^{1/2}$
What are $\partial s/\partial x$ and $\partial s/\partial y$?