Calculus is one of the most widely-used branches of mathematics in economics. Here are two (non-exhaustive) examples of important ways in which economists use calculus:
- to optimize functions. As a simple example, suppose we are thinking about a firm that must choose its price in order to maximize profit. Provided the profit function satisfies a few regularity properties (i.e. is continuous, concave, and smooth), locating the profit maximizing price amounts to identifying the price for which the profit function has zero slope (see figure below). Since we can calculate the slope by differentiating, calculus gives us the means to identify the optimal price.
- to perform 'comparative statics' analysis. In the simple problem above, the firm must choose its price to maximize profit. But often the result of this exercise will depend on a number of other factors. For example, the optimal price will probably depend upon competitive the market is, or how closely substitutable rival products are. A relevant question for economics is therefore "if I make the market more competitive, what will happen to the price?" The typical way of answering this question would be to calculate the optimal price, $p$, as a function of market competitiveness, $h$, and then compute the derivative: $p'(h)$.* Indeed, $p'(h)$ is the answer to the question "how much would $p$ change if I change $h$ a little bit?" Most types of policy involve going into a market and changing something, so the ability to predict what the effects of such a change are likely to be is very useful.
In your principles courses, you have probably seen problems that could be solved in this way, but were instead solved through some other method. almost certainly, the courses were organized like this in order to avoid making calculus a requirement. But, as the problems you tackle become more sophisticated, there comes a point at which the effort needed to learn calculus is much less than the difficulty of trying to figure out a way to do economics without calculus.
To answer your explicit question, there is lots of calculus in economics. But, to answer the implicit question in the background, I don't think you should be too worried by this because:
- Economics involves a lot of fairly easy calculus rather than a little very hard calculus. Primarily, this means calculation of simple derivatives and the occasional bit of integration.
- Doing economics is a great way to become good at calculus! You will get lots of exposure to simple calculus problems. That will give you lots of practice. Moreover, the problems will usually have a fairly concrete application and hopefully make calculus more interesting than trying to learn it from an abstract math book.
- From long personal experience as a student and teacher: it is easier to become good at things you love than it is to learn to love things you are good at.
* If you are wondering why I use $h$ for competitiveness, take a look at the Herfindahl index.