Express dw/dt as function of t, if w = xy, x = cos t, y = sen t, z = t
My first step is to sketch the tree.
w - x - t
w - y - t
w - z - t
dw/dx = y
dw/dy = x
dw/dz = 1
Then:
dx/dt = -sen t
dy/dt = cos t
dz/dt = 1
Then:
-ysent + xcost + 1
Changing the x and y to "cos t and sen t"
Result:
- -sen²t + cos²t + 1
But the teacher's answer is: 1 + cos2t
What's is incorrect?
$\endgroup$ 22 Answers
$\begingroup$Nothing. They used the identity
$$ \cos 2x = \cos^2 x - \sin^2 x $$
to obtain what was written.
$\endgroup$ 1 $\begingroup$$$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}+\frac{\partial w}{\partial z}\frac{dz}{dt}$$ $$=y(-\sin t)+x \cos t + 0\times 1$$ $$=\cos^2 t-\sin^2 t=\cos 2t$$
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