What are the fundamentals of calculus?

Calculus Fundamentals

Calculus is a major branch of mathematics that studies rates of change and accumulation of quantities. It's often referred to as "the study of how things change." The fundamentals of calculus can be broken down into two main branches: differential calculus and integral calculus.

Differential Calculus

Differential calculus is concerned with how things change. It deals with rates of change, slopes of curves, and related concepts. The fundamental concept in differential calculus is the derivative. The derivative of a function at a given point measures how much the output of the function changes in response to a change in its input, at that point. It's essentially the slope of the tangent to the curve at that point. For a function $f(x)$, the derivative is denoted as $f'(x)$ or $\frac{df}{dx}$.

Integral Calculus

Integral calculus, on the other hand, is concerned with accumulation or totality. It deals with summing up infinitely many infinitely small parts to find a whole. The fundamental concept in integral calculus is the integral. The definite integral of a function over an interval is the signed area between the graph of the function and the x-axis over that interval. It's denoted as $\int_a^b f(x) \, dx$.

In the context of this notebook, "Calculus Basics," these are the core concepts you should focus on. They form the foundation upon which more advanced topics in calculus are built.