Chain Rule - Partial Derivatives - dw/dt

Question

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?

Answer 1

Nothing. They used the identity

$$ \cos 2x = \cos^2 x - \sin^2 x $$

to obtain what was written.

Answer 2

$$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}+\frac{\partial w}{\partial z}\frac{dz}{dt}$$ $$=y(-\sin t)+x \cos t + 0\times 1$$ $$=\cos^2 t-\sin^2 t=\cos 2t$$