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The Definition of the Derivative - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

To estimate the derivative of a function, you can use the definition of the derivative: the limit as h goes to 0 of (f(a+h)-f(a))/h. First, evaluate the function at a. Then, you can calculate (f(a+h)-f(a))/h for various values of h, as h gets smaller and smaller. As h gets smaller, it approaches 0 without ever actually being equal to 0. This expression will get closer and closer to some number, which we say is an estimate of the derivative of a function.

Let's do another problem that involves the definition of the derivative. In this problem, I want to estimate the derivative of natural log of x at x equals 2 to the nearest tenth.

Now the way I'm going to do that is, remember that the derivative is actually a limit of this difference quotient. I'm going evaluate the difference quotient for different values of h getting smaller and smaller, and try to figure out what limit those numbers are approaching.

Remember I'm going to evaluate this at x equals 2. That's important. So just going over here. I'm going to make a table of difference quotients. F is my function natural log. When h is 1, this becomes f(3), or natural log 3. F(2) is natural log 2 all over h which is 1. So I need to calculate this on my calculator.

I get 0.4055. Now when h is 0.1, I get f(2.1), natural log of 2.1 minus natural log 2 all over 0.1. Let me evaluate that. I get 0.4879. Then one more. 0.01. This is going to be 0.4987. You can keep doing this for smaller, and smaller values of h. What you will notice is, as h is going to 0, these numbers appear to get closer and closer, to ½. So we would say that the derivative is ½.

So this is the method that you would use if for example you didn't know how to find the derivative algebraically. So I'm using a numerical method here. Just plugging in values for the difference quotient, and letting h get closer and closer to 0. This is how we find the derivative numerically.

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