Find the derivative of the function using the quotient rule
y= (3-(1/x)) / (x+5)
Hello, everystepcalculus.com, this is Tom, and I’ve got everystepphysics.com also. There’s been many searches on the internet for a derivative calculator with steps and that’s exactly what my programs do. That’s what I needed for calculus and that’s what I have programmed for twenty years. Okay, this one is going to be another simple derivatives you gotta know immediately, okay. For instance, if I was to say, what’s the derivative of x to the tenth power, you should right away say 10x to the ninth power. You should automatically know this stuff. Know this in your sleep, okay, because you need to know the basics of calculus or you’ll never pass that class. Even with my programs. You’ll pass it much easier with my programs, but if you don’t know the basics, good luck. So I haven’t programmed really simple derivatives like that. But now we’re getting into quotient rule, and product rule, and implicit differentiation, and things like that so you need a program to do those that’s for sure. And quotient rule is difficult in itself. You have to memorize the formula, you have to be able to put these functions in the formula, and do it all right and somehow get that right on your test. And that’s what we’re interested in is passing tests. We aren’t interested in learning about calculus, we have no interest in calculus but we want to pass our tests. And maybe do some homework, okay. So here’s how my programs work on a derivative regarding quotient rule. We go to my main menu, Index 8, and then I’m gonna go up on the– to get to the “Q” section because “Q” is closer to “W” than the “A”‘s. I’m gonna scroll up here to quotient rule. Choose that, here’s the formula, you write this stuff on your paper so you look like a genius, okay? And we have to enter the function. Well we’re gonna have to press “alpha” in my programs. In every one of my programs you have to press “alpha” first before you enter anything into these entry lines here. Notice the format, you have to add these parenthesis, and then the divide sign, and then the parenthesis, okay. So I’m gonna do that now. I’m gonna add open and closed parenthesis, divided by, open and closed parentheses. Now don’t forget that system, okay. I’m gonna use the slider on the cursor to go back inside the parentheses of the first one. I’m gonna add my function– 3 minus, now we got to open and close parenthesis again, so I add them. Open and closed parentheses, and then use the slider to go back within the parenthesis. And then we got 1 divided by X. Now we’re gonna go inside the denominator parentheses and put in x plus 5. I don’t see the plus sign, I’m gonna go back. Make sure that there’s a plus sign… 5 … very good. I always show you what you’ve entered, you can change it if you want. I say it’s okay, here’s the two functions. And you do your– You start the all the computations, step-by-step. Now remember you can’t do a derivative with the division side, you can’t do an integral with the division sign. You have to change it, there’s always rules in calculus and all the things they do change it to the basics of calculus, which is the derivative you know, multiplying that function times the exponent, subtracting one from the exponent. And then, in integration, adding one to the exponent, dividing by that answer. So know those in it because that’s what every calculation comes down to is to be able to do that, okay. And so we we keep going here, get to the answer here, very good, this is the answer. You write that on your paper, you just got an A on that problem without any effort. Right, if you want to compute that at a point I even have that in here for you. So we’re gonna compute it at– for instance, let’s go, alpha, and put in 3 for x. Again, I show you what you’ve entered in case you made a mistake. Let’s do that. We’re gonna go back and… alpha, let’s put another one in, let’s put in 4. You’re computing it at 4 of x, okay. I say it’s okay. And here’s the– you enter 4 for every x in that derivative. Here’s the prime of x which shows you the derivative, and then we’re entering 4, etc. And here’s the answer: -35 over 1296. Now this is the slope of a line, congratulations. 650 pages in a calculus book to come to this point. A slope of a line. So what do you have here? If you go -35 increments on the y-axis all the way down to -35, and then to the right because it’s positive on the x-axis, 1296.. this is slope. This is rise over run, okay, so you go over 1200 calculate increments all the way on the x-axis, draw a line from that point, through the origin of the graph, and that’s what you found as the slope of a line. That’s what you found the derivative… not tangent yet, because it goes through the origin of the graph. Pretty neat, huh? Everystepcalculus.com, go to my site, subscribe, and enjoy your calculus.