1. ## Partial Derivative

Can someone show me how to get the final answer of cos2t for

Find the derivative of w = xy with respect to t along the path x = cost and y = sint.

I know it's chain rule for partials but i'm still confused on how to obtain that final answer.

2. Edit nevermind, I got it.

$\displaystyle w = xy$

$\displaystyle \frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt}$

$\displaystyle \frac{\partial w}{\partial x} = y$

$\displaystyle \frac{dx}{dt} = -sin(t)$

$\displaystyle \frac{\partial w}{\partial y} = x$

$\displaystyle \frac{dy}{dt} = cos(t)$

sub values in

3. What do you mean? How does that net me the final answer of cos2t?

4. Well, by subbing in those values, we get: $\displaystyle \frac{dw}{dt} = -ysin(t) + xcos(t)$

you defined earlier that $\displaystyle y = sin(t)$ and $\displaystyle x = cos(t)$

so by subbing these to values into $\displaystyle \frac{dw}{dt} = -ysin(t) + xcos(t)$

we get:

$\displaystyle \frac{dw}{dt} = -sin^2(t) + cos^2(t)$

using trigonometric identity $\displaystyle sin^2(t) + cos^2(t) = 1$

we can see that $\displaystyle -sin^2(t) + cos^2(t) = -(1 - cos^2(t)) + cos^2(t) = 2cos^2(t) - 1$

and finally we can use trig identity: $\displaystyle 2cos^2(t) - 1 = cos(2t)$

Hope the more elaborated explanation makes more sense,