## Calculus: Early Transcendentals 8th Edition

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 and to calculate $f_y$, we will differentiate $f(x,y)$ with respect to $y$ keeping $x$ as constant.