Derivatives: Checking your work, how?

[winowmak3r]

Homework Statement

I'm in Pre-Calculus this semester and it's going swimmingly and I thought I'd try and get ahead for Calc I, which I plan on taking this summer. Anyways, all I have really to go off of right now is "How to Ace Calculus: The Streetwise Guide", my brain, and wikipedia. I'm struggling to find more problems to work on (the guide is good in explaining, but I'd like more practice) and I worry that if I just start making up equations to practice finding the derivative that I might end up teaching myself the wrong methods. Here is an example I just came up with off the top of my head. When I tried to check my answer I came up with a few possibilities...

Find Derivative:

f(x) = X^-3

Homework Equations

Power Rule

The Attempt at a Solution

I get:
$$\frac {-3}{x^4}$$

However when I plug it into various "derivative calculators" online I get 0 or some other random solution (one told me the derivative of X=0). Are the calculators wrong? Am I wrong? Is there some way I can check my answers by hand? What I'm doing right now is graphing both the derivative and the original function on my graphing calculator (TI-84 Plus if it makes any difference) and then plugging in a number for X in the equation I got for the derivative and seeing if A) that point indeed exists for that equation and B) if the number agrees with the slope of the original graph. For example, if I plug in x=1 in the derivative I get -3, which is a point on the graph of the derivative and the slope is negative in the original graph( f(x)=x^-3) at the point x=1. I guess my question is am I doing it right? Is there anything I can do to improve my method?

 

[Bohrok]

What I often do is find an approximation for the derivative at a point (or several points) on f(x) and see if it matches f'(x) at those x values.

One good website for math in general is http://www.wolframalpha.com and it can often figure out whatever you're trying to enter in.

 

[Mark44]

Your answer is correct, but you should identify it as the derivative of f. IOW,
f(x) = x^-3 ==> f'(x) = -3x^-4 = -3/x⁴

You're actually using two rules here: the power rule and the constant multiple rule. The latter rule says that if g(x) = k*f(x), then g'(x) = k*f'(x). In words, this rule says that the derivative of a constant times a function is the constant times the derivative of the function.

 

[winowmak3r]

What I often do is find an approximation for the derivative at a point (or several points) on f(x) and see if it matches f'(x) at those x values.

One good website for math in general is http://www.wolframalpha.com and it can often figure out whatever you're trying to enter in.

That website is gold. Thank you! And what you described above about approximating the derivative at several points is what I'm doing now with the graphs, so good to know I'm doing something right then and not teaching myself the wrong way.

And thank you Mark44 for confirming my answer. I actually tried to get a f(x) to show up in Latex but gave up after a while. I never realized I was using the constant multiple rule. I don't think the Guide mentioned it or if it did I missed it. Definitely worth noting.

 

[Agent M27]

I take it you have already learned the limit definition of a derivative, but just incase I will show you how it works. I use it from time to time just to make certain my algebra skills stay sharp and it helps me visualize what's happening on the graph with weird functions. It is a bit lengthy for this one, but it still works all the same. So for f(x)=x$$^{-3}$$ it is as follows in the attached document. I couldn't figure out how to use limits in the post. I hope this helps even though it can be lengthy depending on the problem you wish to differentiate. If you would like I am currently in engineering calc 1, and I can make some copies of my book and email them to you if you would like. I did the same thing you are doing, studying calc while in pre-calc. I found this http://www.karlscalculus.org/" [Broken] quite helpful in my self teachings. Good luck to you.

Joe

 

[winowmak3r]

I'm actually using that site for reference already! Good to know I'm on the right track then.

I did know about the limit definition of a derivative but after learning some of the quicker ways of finding them (like the power rule) I find myself using it a lot less. It is a good idea to use it from time to time just to keep the algebra skills sharp though. I'm always making silly mistakes with the algebra...

Thanks a lot for the attachment too. I found it helpful as I'm prone to making silly mistakes with the algebra and it's nice to see it all worked out. I especially liked the "..now sub in 0 for DeltaX as it no longer poses a danger..." I chuckled. God knows what happens when we divide by zero!

And I would very much appreciate some problems outta your Calc book. That doesn't violate copyright laws does it? If it's legal, PM me them here? Or I could just give you my email.

Thanks again