Basic Rules of Differentiation (Part 3)

In Part 1 and Part 2, we learned the rules of differentiation to handle common functions we might encounter. Now, we will learn about chain rule which will close this chapter of the basic rules of differentiation.

Chain Rule:

Chain rule is perhaps the most powerful of all rules of differentiation. This is due to the fact that many other rules of differentiation are based off it. In order to understand this rule, consider a function f(x) that is a composite function of the functions g(x) and h(x). In this case, the function h(x) is inside the function g(x).

It can be observed that g(x) is the outside function, and h(x) is the inside function. Then, the derivative of f(x) is given by:

This is known as chain rule. It basically says the derivative of f(x) is "the derivative of the outside (keep the inside function the same) times the derivative of the inside."

Examples:

Let's observe some examples of how we can apply chain rule. It is important to understand that the other rules of differentiation still apply.

Once we know how to apply chain rule, we can say that we know how to differentiate. Chain rule is the "pillar" of differentiation. If we understand it, we will be able to learn more complex rules of differentiation and even integration. 

Basic Rules of Differentiation (Part 2)

In Part 1, we learned two basic rules that allow us to differentiate polynomial functions. Now we will learn to deal with functions that involve the multiplication or division of functions. We will learn about product rule and quotient rule.

Product Rule:

Suppose we have a function f(x) that is the result of the multiplication (known as a product) between two functions g(x) and h(x). Then, the derivative of f(x) is given by:

Many teachers and students play with words to memorize this rule. Anyone can make up his/her own way to remember it, but it basically says the derivative of f(x) is "the first times derivative of the last, plus the last times derivative of the first."

Quotient Rule:

Suppose we have a function f(x) that is the result of the division (known as a quotient) of the functions g(x) by h(x). Then, the derivative of f(x) is given by:

In this case, we can say the derivative of f(x) is "the bottom times derivative of the top, minus the top times derivative of the bottom. All divided by the bottom squared."

Examples:

These two rules are quite popular, so let's look at some examples. Notice that we will apply the rules learned in Part 1 when dealing with the terms of the functions.

Now, as seen in the examples, we can differentiate more complicated functions!!!

Basic Rules of Differentiation (Part 1)

There are two basic rules that we can consider the "baby steps" to begin learning calculus. These are the derivative of a constant and the derivative of a variable raised to a constant exponent. 

Derivative of Constant:

The derivative of any constant is zero. Consider the following function f(x), where c is any constant.

Derivative of a Variable Raised to a Constant Exponent:

This is the second rule of differentiation that any calculus student must learn. Consider the following function f(x) where a is any real valued exponent.

In this case, we basically bring down the exponent to the front, and then, subtract 1 to the exponent.

Examples:

Now we will have a few examples that show how important these two rules are. Basically, the derivative of any polynomial function can be obtained from just these two rules.

Many real life situations can be described by polynomial functions, so it is safe to say that these two rules of differentiation will be used frequently.

The Relationship among Position, Velocity and Acceleration

Position, velocity and acceleration are concepts all of us deal with every day. We give directions to a friend about our location in town in order to meet up at a restaurant. We drive to work as we push the acceleration and brake pedals in order to change the velocity of the car. These are just every day examples of how these concepts affect our life. We do not give much thought to these actions, but at the mathematical level there is more than just doing our daily activities.

The relationship among position, velocity and acceleration is one of the fundamental applications of calculus. From the point of view of calculus, these quantities are related by differentiation and integration. This tells us that if we know at least one of them, we can calculate the other two. The figure below shows the relationship among these quantities.

The arrows in the figure above show the directions in which we have to either differentiate or integrate in order to obtain the other quantities. A simple example is shown next.

Solution:

Differentiate the position equation to obtain the velocity. Then evaluate at t = 10.

Differentiate the velocity equation to obtain the acceleration. Then evaluate at t = 10.

At t = 10, the velocity and acceleration are 97 and 10, respectively, in their corresponding units.