Calculus through Data & Modeling: Differentiation Rules

Derivatives of Polynomial, Exponential, and Logarithmic Functions

In previous course, we defined and calculated the derivative as a limit In this module, we will examine the derivatives of some important functions, including polynomials, exponentials, logarithms, and trigonometric functions We will also learn differentiation rules which will help us to compute derivatives more efficiently. Finally, we will generalize the idea of a derivative to multivariable functions, and learn how to find derivatives and rates of change on a graph in space

The Product and Quotient Rules

The formulas of this section enable us to differentiate new functions formed from old functions by multiplication or division

Derivatives of Trigonometric Functions

Before starting this module, please review trigonometric functions, in particular their graphs In this module, we will develop formulas to find derivatives for the common trigonometric functions of sine and cosine Together with the product and quotient rules, the derivatives for the remaining trigonometric functions are formulated These new derivative formulas are then added to our catalog to use and apply to solve problems related to rates of change

The Chain Rule

Many functions are created through composition of other functions In this module, one of the most important of the differentiation rules of this course is developed which will allow us to find derivatives of the compositions of functions This rule is called the chain rule and has a variety of applications

Partial Derivatives

In this module, the notion of the derivative is applied to multivariable functions through the notion of partial derivatives Algebraic rules are developed to find partial derivatives of multivariable functions as well as their geometric interpretations The development of the tools of calculus to multivariable functions allows for further analysis of more complicated data sets

Directional Derivatives and Gradient Vectors

In this module, we continue the application of partial derivatives to find rates of changes in any direction by developing the theory of directional derivatives and gradient vectors These new tools of multivariable calculus can then be applied to problems in economics, physics, biology, and data science

Final Project: Flight Path

Apply the theory of this course to model a flight path for a landing aircraft