Find the derivative of the function using the quotient rule
y= (3-(1/x)) / (x+5)
Derivative of ln[x^(2/3)]
Hello, everyone. This is Tom from everystepcalculus.com and everystepphysics.com. In this video,how to do the derivative of the log, natural log of some function. And you can see function on your screen. Let’s do it. Index 8 to get to my menu. And we’re already at log of x derivative. Scroll down to that. Press enter. And we’re going to differentiate it. Because that’s what we’re doing. We’re finding the derivative. And we enter our function. In this case, you have to press Alpha before you enter anything into these entry lines on my programs. I’m going to press alpha and then it’s second to get to the log function. Here’s log of our problem. And it’s X to the two-thirds. Close off the parentheses. I always show you what you’ve entered, you can change it if you want. I say it’s okay. You press number one. Here’s the formula. Here’s over after the derivative of this. U prime over U is the formula, etc. Pretty simple but to remember all these different formulas and rules in Calculus is very difficult. That’s the reason I needed to program this stuff. The answer is 2 over 3x. Pretty neat, huh? everystepcalculus.com. Go to my site and subscribe so you can see more videos that I made you. Have a good one.
Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. I’m gonna do a a definition of the derivative or difference quotient problem right now. et’s do it. Index 8 to get to my menu. Scroll down to definition of a derivative. There it is there. I gave you a little bit of help in case your professor does some tricks change of X instead H. There’s all kind of tricks and this is one of the most nonsensical things I’ve ever seen in calculus. It’s all nonsense but.. I’m going to choose number two, here. For a square root function. I’m going to enter the function. You have to press Alpha before you enter anything into these entry lines. Alpha, going to use the square root second square root of X minus 4, close off the parentheses. And I say it’s okay I give you a chance to change it in case you make a mistake. Here’s the formula. And remember, you’re replacing X plus H for every X in the function. And then in the case of square roots, you’re using the difference squares to get rid of the radical signs. So we’re multiplying it times the plus of the same thing which makes those squares. I’ll show you that in the next one, here. So they become squares and when you square a square root, you get what’s inside the square root. Here it is. And in the bottom, of course H, you do the same thing because they’re really taking the numerator and denominator exactly the same thing which you have to do So you don’t change the function. And then on the top, we multiply it all together. And you come up with H on the top, these H’s cancel. Becomes one and now we’re gonna when H equals 0, then we apply that to this down here which cleans that up and makes it the same thing. Two of each right here. So that’s where the two comes into it. Here’s the answer one over two times square root of x minus 4. As usual, calculus makes a big big deal about this when
An email from a Calculus student:
First, I want to sincerely thank you for the support of your wonderful programs. You’ve inspired a knowledge of calculus that my prof cannot.
I’ve got a problem to find the extremas of an exponential function over a given closed interval but don’t know which program to use. Here’s the problems:
f(x) = (3x-1)e^(-x), on the interval [0, 2]
f(x) = (ln (x+1))/(x+1), on the interval [0, 2]
any advice would be greatly appreciated!
In short you enter the function into my “graph by hand” menu program.
Calculus to me is like teaching you how to multiply through the 9’s and then make a student take three semesters siting a million areas of the usage of multiplication.
Calculus finds two things, the derivative finds the “slope of a line”, that’s it!!! and the integral finds the “area” under a smooth curve (x^2,x^3,x^4,x^5 etc and sin(x) and cos(x) (called a sine wave) as you graph those on an x y graph, and only finds the area, if you give that integral a range, called a “definite integral” an indefinite integral finds nothing.
To me calculus is the Sudoku of math, the study of cross word problems. Something to do while waiting for a plane to Phoenix. In calculus they don’t say “rose” (ya know the flower) they say “hibiscus mutabilis” they constantly make easy things into extremely hard things. Linear approximation is a prime example, as well as related rates.
That said, “extrema” “max and mins” of a function is the absolute or maximum high points or low points as you look at a graph of a function. If you graph -x^2 (minus) this is a smooth mountain, extrema is standing at the top. The opposite is true of x^2 (positive) this is a smooth valley, if you graph it, and you are under that valley, touching it with your finger at the lowest point.
That said, it just so happens that when the slope of line is “horizontal” it has a slope of zero, and if you set that horizontal line on top of the mountain it will touch at only one point (tangent) and that point will be the highest possible point on that mountain. So calculus take the first derivative of a function (slope of a line), sets it equal to zero and then solves for the x values. It then puts those x values back into the origininal function, and when solved, finds the “y” value. That point (x,y) is the maximum or extrema of that graphed function. Those x values are called “critical numbers” because they lie on the x axis. They become “critical points” when you plug them into the original function and solve for “y”.
Now to me in your first example, given what I’ve just taught you, They say “over an interval” and then give you the inteval [0,2] (Notice the brackets which indicate an interval where parenthesis would indicate a point (x,y) in math — There is only one critical point (no interval) where the hoizontal line would touch the graph (tangent), so I guess this “over the interval” is to throw you off, or check whether your understanding is as good as mine.
My graph program will do all this for you, but in the first example if written properly (get used to doing this) would look like this: (3*x-1)*e^(-x). Notice the times sign in front of the “e” that tells you product rule to find the derivative. In the second example quotient rule would be used to find the derivative.
Thanks, for the kudos, Tom
This is a pre calculus help video on derivatives calculus, actually a derivatives calculus program app for the ti 89 titanium calculator. To get to my main menu. You press 2nd alpha to enter the letters I n d e x and then press alpha again to enter the eight and the open, and closed parenthesis. Press ENTER and you’re into my menu. You scroll to where your choice is, right now you want to know how to do derivatives calculus. So we’re going to scroll to e to the x derivatives calculus in my menu. Press ENTER and we’re going to enter our function. You have to Press alpha before entering any variables in my programs we’re going to enter this example up here six times x squared. I show you what you’ve entered you can change it if you want. I say it’s ok. Press enter and you write everything on your paper to get the step by step calculations of derivatives calculus. I show you the formula, you have u here, d u equals the derivative twelve x format of putting it in, is like this, you substitute six x squared for u and the answer is twelve x time e to the six x squared now this d u, you multiply it times anything in front of the e to the x in case you enter it that way or have that problem. Pretty neat huh? Everystepcalculus.com. Go to my site, enjoy my programs, and pass calculus.
This is a TI-89 step-by-step program app for the derivative of sin2x or some function within the parentheses
to get to my main menu you press second alpha to enter the letters i_n_d_e_x and then press alpha again to enter the 8 and the open and closed parentheses press enter and your into my menu you scroll to where your choice is in this calculus calculator program app i’m demonstrating how to solve step-by-step the derivative of sin2x you can see the choices here of on some my menu many many things projecting A on B quadratic formula, quotient rule relative extrema, that’s graphing by hand but we’re going to do the derivative of sin2x here so we go down to sine of A to the X derivative and press enter you can do cosine or sine we’re going to enter the function you have to press alpha before you enter anything in my programs and so we’re gonna do sine of five times x press enter. I show you what you’ve entered corrected if you want to say it’s okay. I give you the chain rule formula you write this stuff all in your paper to get the step-by-step calculations U equals five x etcetera derivative of f of u sine of x cosine of u and so you have your answer here five times cosine of five x pretty neat huh? EveryStepCalculus.com pass calculus and enjoy my programs.
This is a video from everystepcalculus.com demonstrating how my programs work on a TI-89 Titanium calculator and other calculators in the TI system for calculus and physics problems ok the derivative of log of x and let’s get started to enter my menu you have to push second alpha and then put i n d e x in here and then press alpha to put the eight and the open closed parenthesis and press enter and you’re into my menu. I’ve already scrolled down to log of x it’s all alphabetical. I’m going to press enter you want to differentiate it and we’re going to enter you have to press alpha before you enter anything in these entry lines of my programs. So I’m going to press alpha I’m gonna put log and we wanna put this time cosine of five times x close off the parenthesis twice there. I always show you what you’ve entered so you can change it in case you made a mistake. I say it’s ok and you notice here’s the formula u prime over u u is what’s ever inside the parenthesis of the log natural log and so we have the cosine of five x the derivative of cosine of five x is minus five sine of five x and then u prime over u which turns out to be sine over cosine which is really tangent so the answer is minus five times tangent of five x pretty neat huh
everystepcalculus.com. Go to my site buy my programs and pass calculus.
This is a video from everystepcalculus.com demonstrating how my programs work on a TI-89 Titanium calculator and other calculators in the TI system for calculus and physics problems ok derivative of log of x and let’s get started to get to my menu you have to press second alpha to put in i n d e x in here the letters and then press alpha to put the eight and the open closed parenthesis. Press enter and you’re into my menu I’ve already scrolled down to log of x natural log of x these are all alphabetical and we’re going to press enter and we’re going to differentiate log of x ahh you have to press alpha before you push put anything into these entry lines of my programs. So you have to remember that and this is a tricky ahh supposedly simple problem that I’ve seen on tests so. Let’s do it. I’m going to press alpha. I’m going to press second to get to the log and then the’ve got second again to put the log in there and then we’re going to put five times x and double parenthesis of course I always show you what I’ve entered what you’ve entered so you can change it if you want. I say it’s ok you’ll notice the formula you prime of u u is always in the parenthesis here in this case log of five x and u prime is one over x and so your answer is one divided by x times log of five time x. Pretty neat, huh? EveryStepCalculus.com, go to my programs buy my programs and enjoy passing calculus.
Summary of Video
How the TI-89 calculator solves the derivative of sin2x:
~ Bring up the menu and scroll to sin of a of x
~ Enter functions by pressing alpha before
~ Program will give you chain rule formula
~ Programs Produces answer all step by step
This is a t i eighty nine titanium step-by-step program app for the derivative of cosine of two _x for some function within the parenthesis there to get to my main menu you have to press second alpha to enter the letters i_n_d_e_x and then press alpha again to enter the eight and the left and right parenthesis press enter into my menu you have many choices in this menu of course you can scroll way down all alphabetical equation of a tangent line eliminate the parameter, dot product, cross product, whatever you need here all step-by-step all perfect and we want to do cosine derivative scroll to that uh… you have to press alpha before you enter anything in my programs and so we’re going to enter cosine and let’s enter rather than two _x let’s enter _x squared. I show you what you’ve entered so you can change it and i say it’s okay give you the chain rule formula, you write all of the things down step-by-step calculations on each screen. So you can get an _a in calculus uh… derivative of cosine of _u is minus sine of _u and here’s your answer two times _x minus sine of _x squared. Pretty neat huh? EveryStepCalculus.com, go to my programs. Enjoy my programs, and pass calculus.
This is a video on the definition of the derivative also called a quotient, difference quotient. A very difficult program wouldn’t be difficult for you and me, if you have all day, all night and several days to study it and get the handle on it and really think about it). But this program may help in your calculus endeavors. Gonna get started, press 2nd Alpha (this shows that you can enter letters in the calculator). My code for getting into my program is index then you press alpha again to get number 8 and the (). Should be index8 (). Press enter and that takes you to the menu, where you can scroll down and select what you want. We’re working with definition of derivative here, select and press enter and put in formula or function. Press Alpha first then enter everything in the boxes on the program. We’re gonna do 3*x^2-x+1 go to ok or change it if you want. Write down the formula that comes up. First thing to do when writing a problem is to write the formula down and then you start substituting x+h for every x in the function, which I’ve done here. 3*”x+h” or parentheses actually, squared, minus “x+h”+1 and then the original formula all over h.
F’ (x) =lim
Everystepcalculus.com, enjoy my program visit my website.
The equation of a line to a point on a curve (point slope form) includes the slope and the position of that line on that curve function. It’s better than the derivative because the derivative only tells us the slope. Again in Algebra the professor forgot to tell us the importance of that and the relationship to the derivative. Didn’t make it interesting enough to sink in and how it relates to the real world.
You have a function. Has to have x^2 in it to be a curve from my understanding, Example y or f(x) = 3x^2
Graph that and you have some form of curve in this case a “valley” parabola, (my own word), -3x^2 and you have a “mountain” parabola (again my own word).
Pick any point “(x,y)” Example: (3,12)
Point = (3,12)
x = 3
y = 12
Find the derivative: f(x) = 3x^2
f’(x) = (2)(3)x^(2-1)
Compute the derivative at the point “x”
f’(3) = 6(3)
= 18 = m = slope
Point slope form = y = mx + b
y = 12 so:
12 = mx + b
m = 18
12 = 18x + b
x = 3 so:
12 = 18(3) + b
= 54 + b
b = 12 – 54
= – 42
y = mx + b
= 18x + -42
If you graph this equation along with the original function you’ll see the tangent line to that point on the curve
The slope = 18/1 (rise over run)
The angle of that tangent line = tan^(-1)(18/1) = 86.8 degrees
(make sure your calculator mode is in APPROXIMATE and DEGREES)
Fabulous and exciting, right? lol Tom
p.s. You’ll love my programs
Have a test or quiz on point slope form? Here is a video example using the programs on the TI-89 calculator: (Click Links below)
Ask anybody “what is a derivative?” and you’ll quickly find out that nobody can tell you exactly what it is in no uncertain terms without any question. If fact, most people you ask who have taken calculus can’t even come close. Think I’m wrong, try it and you’ll find out what I did, it took me eight years after college to find out and even then not exactly. That’s pathetic and unacceptable in my opinion. Of course I was programming the TI Calculators in calculus at the time so I had some interest to even ask the question. I mean, who cares right? No one out of college will ever calculate a derivative or integral again in any job outside of re-teaching it as a professor, so I/we understand that.
I had some interest because of programming step by step calculus into the calculators and while teaching tennis to this guy named Mike, I find out – he at one time worked at NASA. At the time of me teaching him tennis he had left NASA was a professional black jack player. Went all over the world making money at black jack in the casinos. I was also fooling around with on line poker at the time and asked him how come black jack and not poker? He said poker was too much gambling and black jack is relatively sure. He said he used differential equations to help him count cards and change the odds in his favor. Anyway I asked him what was a derivative? He said immediately that it was the slope of a tangent line to a curve. I said “but when I graph the derivative there is a line – but no tangent line and the slope is off.” He said the graph of the tangent line is meaningless, “of no value”. He said when you compute the derivative of a function at an “x” value you come up with a number and that is the slope of the tangent line at that point on the function. I thought even that was fabulous and an eye opener, but we had finished picking up tennis balls and so I let it go and started to again teach him tennis.
After that moment, I kept thinking and thinking and thinking of what he said and then it dawned on me. The number you get when computing the derivative at the chosen point “x”, no matter how deep the original function is the numerator of a fraction with the denominator = to 1. That was rise over run. If the numerator is 12, that is 12/1. You go 12 spaces up the “y” axis and one space over on the “x” axis. Draw a line from that point through the origin of a graph (0,0) and that is the exact angle or slope of the line of the derivative. For that line to me tangent, it must touch the curve at only one minute point, so that line has to be transposed to do that, however it will still be the exact slope. To transpose that line you have to compute the equation of that line to the desired point (point slope form) and then graph that function and you have the perfect picture of what a derivative is. That’s fabulous. (Equation of a line at a point on a curve is in my calc1 programs). Now at the next party you know all there is of “what’s a derivative” and can look like a nurd, I mean nerd. lol, Tom
p.s. (The exact angle of that slope by the way is tan^(-1)(12,1), Fabulous!!
In my first day of calculus I – the chalk was flying. That professor started with the fundamental theorem of calculus, probably said “if it exists” a hundred times, and never let up from then on. It got worse from then on and never better. I, with others – I’m sure – sat there in disbelief and in a fog. I was 50 years old that day. I remember 3 semesters later I turn to the guy next to me and ask, “What the hell is a derivative?” He whispers, “I think it’s an angle of a line or something”.
Try it yourself go up to any one of your friends, ask them first if they’ve taken calculus and then ask them what a derivative is, and see what the say or don’t say. That’s the way all of my professors taught in my experience of college, I never enjoyed one class. If it was me – and my inability to learn in a class – that kid would have told me what a derivative was without any question as well as others in the class, however nobody knew (I asked several others after that kid) they didn’t know either. We were all three semesters into calculus and didn’t know what a derivative was. What the hell is that? Isn’t knowing what a derivative is, in no uncertain terms, more important than the fundamental theorem of calculus???