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The Definition of the Derivative - Problem 2
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.