Computing derivatives?

Question

Let $u: \Bbb R^2\to\Bbb R$ and define $f(t)=u(\sin(t),\cos(t))$

How may I calculate $f'(t)$?

Note: I am expecting the answer to include symbols like ${\partial u}/ {\partial x}$

Is there a law for such kind of derivatives?

Answer 1

Whenever you're not sure about how to compute a derivative, the best approach is to break it down into a composition of functions and apply the chain rule.

Define $g(t) = (\sin t, \cos t)$. Then $f(t) = u(g(t))$, i.e. $f = u \circ g$. Thus $$ Df(t) = Du (g(t)) \cdot Dg(t) = \begin{bmatrix}\partial_x u & \partial_y u \end{bmatrix} \cdot \begin{bmatrix}\cos t \\ -\sin t \end{bmatrix} = \cos (t) \, \partial_x u (g(t)) - \sin(t)\, \partial_y u (g(t)) $$ So $$ f'(t) = \cos(t) \, \partial_x u(\sin t, \cos t) - \sin(t)\, \partial_y u(\sin t, \cos t) $$