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Derivatives of Logarithmic Functions - Concept

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.

The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

Talking about derivatives of functions and one special function is the natural log function. Now recall the definition natural log y=lnx means e to the y equals x. And remember that e is that special number 2.71828 itÂ’s an irrational number and has an infinite non repeating decimal representation just like pi. Treated that it's a very important number though because when you create andexponential function e to the x its derivative is also e to the x. So thatÂ’s one of the important things about this number.
Now, I've graphed y equals e to the x and I've graphed y equals lnx and remember these are inverse functions of one another and so they have the symmetry about the diagonal line y equals x. Now I want to talk about the derivative of natural log so I start with the definition of derivative limit as h approaches zero f of x plus h minus f of x over h. And then I substitute in natural log for f and this is what I get. Now this is, our goal here is to try to figure out what the derivative is and we're not going to be able to do it with Algebra. So instead what I'm going to do is play a little game of guess the function, now the way I'm going to do that is I need to get this in a form that I can actually graph on my calculator. And so what I'm going to do is approximate this limit by taking an h value that's really really small like 0.001 and I'm going to approximate this limit with natural log of x plus 0.001 minus lnx all over 0.001 and that will be pretty close, close enough for our purposes to tell what function the derivative of lnx actually is. So thatÂ’s my goal for this lesson to graph f of x equals lnx and it's derivative on the ti84 and identify the derivative function.
LetÂ’s do that now, so we're looking at the ti84 first thing I want to do is go into graph mode so hit the y equals button and I'm going to enter my function natural log x. Now for y2 I need to enter the derivative or my approximation for the derivative which is parenthesis natural log of x+0.001 minus natural log of x. Close parenthesis and then divide that by 0.001. So this is a difference quotient with h equal 0.001 it'll be pretty close to the limit as h approaches zero of the difference quotient.
Let's see what the graph gives us, okay so you're looking here at first of all this upward this increasing curve thatÂ’s the natural log curve. This decreasing curve here is its derivative, and it kind of makes sense that the derivative should be decreasing because remember gives me slope of natural log at any point. Here very close to, when x is close to zero the slope is steeper and so the derivative should be greater. And over here where this slope is less steep the derivative should be less great and thatÂ’s what happens down here. Let's try to figure out what this function as the derivative function so I'm going to hit the trace button. Now right now it starts with x equals zero and of course both these functions are undefined x equals zero let me cursor to the right a little bit.
Okay now we're on one of the curves, weÂ’re actually on the natural log curve so you need to hit the up and down button to switch to the other curve and now I'm on my approximation of the derivative. Now x equals 0.1063 this is not going to be very helpful to me, let's actually advance to x equals 0.5 and I can actually type 0.5 in. Now notice what the y value is, it is really really close to 2, let's try x equals 1 and I can just type 1 very close to 1. Let me type 2 notice that this value is very close to a half. And so I'm starting to notice the pattern that the y value is close to the reciprocal of the x value. Let's try 3, 0.333 definitely close to one third and then I'll try one more 4, very close to a quarter which is 0.25 so I hypothesize that the derivative of this function is y equals 1 over x. Let's take this back to the board.
Alright so what we just figured out is that the derivative of natural log is actually the function 1 over x. ThatÂ’s another very special thing about the number e because this is the logarithm of base it's got a very nice Algebraic derivative and so really important result and we'll be using this a lot in upcoming problems is that the derivative of lnx is 1 over x.

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