Answer
a) See the explanation below.
b) See the explanation below.
c) See the explanation below.
Work Step by Step
a) The partial derivatives for $f_x(a,b)$ and $f_y(a,b)$ are described as:
$f_x(a,b)=\lim\limits_{l \to 0}\dfrac{f(a+l,b)-f(a,b)}{l}$
and
$f_y(a,b)=\lim\limits_{l \to 0}\dfrac{f(a,b+l)-f(a,b)}{l}$
b)
Suppose we consider $f(a,b)=c$, then the point $ p(a,b,c)$ will be a point on the surface $S$ such as $z=f(x,y)$.The partial derivatives for $f_x(a,b)$ can be interpreted as the slope of the tangent line to the curve at point $ p(a,b,c)$ represented as the intersection of the surface and the plane $y=b$ , that is, parallel to the $xz$-plane along the surface $S$.
On the other hand, the partial derivatives for $f_y(a,b)$ can be interpreted as the slope of the tangent line to the curve at point $ p(a,b,c)$ represented as the intersection of the surface and the plane $x=a$ , that is, parallel to the $yz$-plane along the surface $S$.
The first partial derivatives for $f_x(a,b)$ is the rate of change of $z$ with respect to $x$ keeping $y$ as constant or fixed in accordance to the variable $x$.
On the other hand, the first partial derivatives for $f_y(a,b)$ is the rate of change of $z$ with respect to $y$ by keeping $y$ as constant or fixed in accordance to the variable $y$.
c)
To calculate $f_x$, we will differentiate $f(x,y)$ with respect to $x$ keeping $y$ as constant or constant on the other hand, to calculate $f_y$, we will differentiate $f(x,y)$ with respect to $y$ keeping $x$ as constant or constant.
