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. And so I'm starting to notice the pattern that the y value is close to the reciprocal of the x value. Let me type 2 notice that this value is very close to a half. 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. 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. 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 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.
Let's try to figure out what this function as the derivative function so I'm going to hit the trace button. And over here where this slope is less steep the derivative should be less great and thatÂ’s what happens down here. Here very close to, when x is close to zero the slope is steeper and so the derivative should be greater. 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. 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. 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. Close parenthesis and then divide that by 0.001. 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. 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. 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.
WHAT IS THE DERIVATIVE OF LOG X PLUS
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 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. 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. And then I substitute in natural log for f and this is what I get. 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. 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. So thatÂ’s one of the important things about this number. 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. 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. Now recall the definition natural log y=lnx means e to the y equals x. Talking about derivatives of functions and one special function is the natural log function.