Derivative of a multivariable function at a point

Question

I'm facing this problem of finding the derivative of a function $f(x,y)= (\sin^2 x \cdot \cos y, xy)$ at the point $(\pi,\pi/2).$ The problem is that I don't know if I should calculate the partial derivative and then plug in the points or find the directional derivative or what exactly because it's the first time to find such function with two components.

Answer 1

You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .

$$f_1(x,y)=sin^2(x)cos(y)$$

$$f_2(x,y)=xy$$

$\frac {\partial f_1} {\partial x}=cos(y)2sin(x)cos(x), \frac {\partial f_1} {\partial y}=-sin^2(x)sin(y)$

$\frac {\partial f_2} {\partial x}=y, \frac {\partial f_2} {\partial y}=x$.

So the derivative is a $2x2$ matrix $D_f(v)=\frac {\partial f_i} {\partial x_j}$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)