Blog post #3: Professor Kate Cornman

Hello! My name is Professor Kate Cornman and today I will be teaching you about derivatives. When I was first learning derivatives I had a tough time. I understood how to find it but I did not understand why we even needed them. So during this class we will cover the basis of derivatives such as why we take the derivative and the process behind it. I will briefly cover the basic derivative rules but the focus of this class is to examine the reasoning for derivatives.

But first, Congratulations! We have traveled back in time to the 2012 Summer Olympics in London. The atmosphere is electrifying and it’s time for the track and field events.  The infamous Jamaican man, Usain Bolt, is preparing for his next race the 100m sprint.

In less than a minute,  the world is changed. Usain Bolt wins the race with in 9.58 seconds.

Now think about his average speed, it is 10.4 meters/second or about 23.5 miles/hour.

We just find the change in distance/ change in time, or find the slope of the graph.

Anyways, we all know he is really fast but he is not just going to take off and run 10.4 meters/second right off the sound of the gun, obviously, at first he is going to start out going a little slower.

The biggest question is : How fast is Usain going right now at a particular time? This is not the average.

Think about:  WHAT’S HAPPENING IN THIS INSTANT?!

Well this relates to Instantaneous Rate of Change! You find this by finding the average rate of change in smaller and smaller numbers approaching zero.

When you research Bolt’s fastest instantaneous rate of change, its actually close to 30 miles per hour!

Without this instantaneous rate of change, we would not get the full picture.

But how did they find this out, though?

My thoughts actually. Well, class we will examine a very long formula which will explain the background to derivatives.

Difference Quotient

f(a+h)- f(a)

(a+h)- a

This long and annoying formula explains that you need to find the Average Rate of Change for smaller and smaller values until you get closer and closer approximations of the derivative.

This is great way to start learning about derivatives because if you were like me and in high school your teacher just told you we are learning derivatives today and this is how you find them, how will you form the basic idea of slope and instantaneous rate of change?

That is why I am here today, everyone.

For example, estimate the derivative of f(x)= 5xat x=0

Remember Difference Quotient is f(a+h)- f(a)

(a+h)-(a)

1. h1=.1 =     f(-1)- f(0) =   f(a+ h1)- f(2) = 5.1- 50/ .1-0 = 1,7461

1-0 (a+h1)-a

II. h2= .001  = 5.001- 50/ .0001-0 = 1.6107

Now, quickly I will reveal to you the real derivative:

f’(x)= ln (5) 5x= f’(0)= 1,609

See if you continue to find smaller and smaller numbers such as .00001 and use the difference quotient then you can manually find the derivative.

But you are probably asking yourself how did I find the derivative?

Well, let’s get to it.

Power Rule:

For constant real # n :

d xn= nxn-1

dy

Chain Rule:

d [ f (g (x) ] = f’ (g (x)) . g’ (x)

dx

Product Rule:

(f(x) . g(x)= f’ (x) . g(x) + f(x) . g’(x)

Quotient Rule:

f(x) = f’ (x) . g(x) - f(x) . g’ (x)

g(x) g (x)2

Although, I did not spend as much time on derivative rules as needed, what I wanted the class to understand today was the background on derivatives.