Why is the derivative important?

The derivative has many important applications both from elementary calculus, to multivariate calculus, and far beyond.

The derivative does explain the instantaneous rate of change, but further derivatives can tell the acceleration amongst other things.

With optimization, the derivative can tell us where the best place to sit in a room is, if the room is filling up with smoke, and at what time it is the best to sit there. The derivative can help with many optimization problems.

The partial derivative tells us the direction of variables at a given time and the total derivative tells us where the slope increases the most and where. This is one way we can optimize in $\mathbb{R}^3$. The derivative can be applied to water flow and generally tells us much about how things change with respect to another variable.

The derivative further can help in industry with economics, healthcare, engineering (especially), and many other things. Business has many applications as well. Your professor might not have time to delve into these applications as much as you would like because it is a calculus class, not an "application of the derivative" class. Although, he should definitely discuss these issues at some point. I have had some professors in my time who glossed over such subjects, but in multivariable calculus, they go way more in depth with them. I don't suggest switching your major without speaking directly with your professor about your difficulties.

If you have further questions, I encourage you to ask your professor in office hours the same exact question and voice your concerns there. A good professor will encourage and motivate your learning outside of the classroom if you show initiative and ask.

I'm going to take a slightly different tack than most of the other answers here and point out that "important" (the word used in the title of the question) and "useful" (the word used in the body) are not exactly synonyms. Something can be important in different ways:

• It may be important because it is useful for solving real-word, practical problems
• It may be important because it is useful for solving theoretical, non-applied problems
• It may be important for historical reasons
• It may be important because it is surprising or counterintuitive
• It may be important because it illuminates a mystery

A lot of these might be summarized by saying that something is "interesting". I think of "interesting" and "useful" as orthogonal axes of value, in the sense that they are two completely independent ways of saying why something is worth knowing.

Most mathematicians, I would venture to say, are motivated by things other than "practical applications". (Some even actively scorn applications, although I do not go that far.) According to tradition, when Euclid was asked "Why is this useful?", he replied sarcastically

"Give him threepence [lit: a three-obol piece], since he must make gain out of what he learns."

The derivative -- and calculus in general -- is important and interesting in many of the above senses, quite apart from the practical applications (which, it must be said, are extremely abundant). From the days of the ancient Greeks to the time of Newton and Leibniz, philosophers struggled to understand the nature of motion itself, which many of them regarded as fundamentally paradoxical: if, in any given moment, zero time elapses -- and therefore in any given moment an object's position does not change -- how is motion possible? More generally, how do we get the experience of smooth, continuous change from a sequence of infinitely many distinct points in time? (The geometric version of this is: if a point has zero size, and a line is just a set of points, how do lines have size?)

People have thought of these questions as interesting for centuries because they are head-scratchers. They lead one to ponder the infinitely many, the infinitely small, and the ways in which infinities can balance each other out to produce finite quantities. That stuff is cool, and Calculus provides a set of techniques for figuring it out, and a language for talking about it coherently.

Once you formalize it, derivatives also allow you to discover things that are genuinely surprising, for example that it's possible to have a curve that is continuous everywhere but not differentiable anywhere. (That statement doesn't even make sense without derivatives, but if you understand derivatives you can begin to appreciate how absolutely baffling and nonintuitive such a thing must be.) The capacity to be surprised and the ability to contemplate the infinite are part of our sense of wonder, and essential components of being human. I would say that's pretty important, whether or not it's "useful".

The derivative locally measures how much a function stretches its domain at a point. If it is negative, there is both stretching and reversal of direction.