We have finally started taking partial derivatives in Calculus IV. This is definitely one of my favorite parts of the entire Calculus sequence. Today, we finished up talking about limits of functions of several variables and the concept of continuity. We, then, introduced the concept of holding one variable fixed and finding the derivative of the function with respect to the remaining variable. After seeing the picture drawn on the board of two curves, or slices of a surface, and identifying the partial derivatives as the “slope” of tangent lines to those two curves, one of the students made a keen observation: Can’t we have infinitely many partial derivatives by going in any direction, not just the x or the y direction? Answer: yes, but . . .
We will be covering directional derivatives in couple of sections down the road, and we’ll see that even for those cases we only need the two directions to find all the rest. Next time, we’ll cover linearization, tangent planes, and differentials. Another one of my favorite topics is coming up: the Implicit Differentiation Theorem.
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