# Need help with chain rule for relating ds/dt to dx/dt and dy/dt

by rectifryer
Tags: chain, ds or dt, dx or dt, dy or dt, relating, rule
 P: 10 1. The problem statement, all variables and given/known data s=$\sqrt{(3x^2)+(6y^2)}$ 2. Relevant equations None 3. 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.
 P: 294 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?
 Admin P: 21,585 Well the relevant equation under 2. Relevant 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....e.4/index.html http://www.math.ucdavis.edu/~kouba/C...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.
P: 10

## Need help with chain rule for relating ds/dt to dx/dt and dy/dt

 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?

 Math Emeritus Sci Advisor Thanks PF Gold P: 38,706 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 $\partial s/\partial y$?