# Lesson Series

Differential calculus is the study of the slopes of functions and their application to real world problems. The slope of a function $f(x)$ at a specific point is called a derivative and is denoted by two common notations: $f'(x)$ and $\frac{df}{dx}$. We provide a listing of real world examples where derivatives are used to describe real world phenomenon.

Here we show that a derivative is a function limit. We provide some simple examples of how the function limit definition is used.

In this lesson we derive the formula for the tangent line given that we know the function and its derivative.

We discuss here how a function is continuous at a point if it is differentiable at that point. However, the converse is not true. If a function is continuous at a point it is not guaranteed that it will be differentiable there.

The Chain Rule shows how to find the derivative of a function composition. The Chain Rule states that \[\frac{d}{dx}\left[f\circ g\right] =\frac{df}{dg}\cdot\frac{dg}{dx}.\]

We present here the most important differentiation formulas. We include derivatives of $x^n$, the product rule, the quotient rule, and the chain rule.

We explain how the derivative is the rate of change of one variable in relation to another.

The Leibniz notation is where we denote a function's derivative by $\frac{df}{dx}$. We provide an explanation of where the Leibniz notation comes from.

We motivate the fact that we can find as many other derivatives of a function as possible and emphasize the use of the Leibniz notation to describe them. We calculate all of the derivatives of $f(x)=x^2$ and $f(x)=e^x$. We provide links to lessons that detail the meaning of the second derivative.

An antiderivative of the function $f(x)$ is another function $F(x)$ where $F\,'(x)=f(x)$.