1. ## Test Time

Hey,
I took a test this Monday on Derivatives, and ran into a problem. I didn't have enough time to finish the test. We got the test back today, and I got all of my answer right, except I didn't get to the rest of the test, and ended up getting a 36%. Supposedly a few others are having the same problem, and he is letting the class re-take the test (different form though), except he is giving us 2 hours instead of 1 and a half. Now, I know how to find derivatives and have done plenty of homework. My problem is taking forever to complete them. Would anyone happen to have any tips or tricks at finding the derivative of a problem quickly? Even with the different rules for making the derivative easier to find, it still takes a while.

Thanks!

2. TKHunny's off-the-wall test-taking hints...

1) Never, ever stew over a difficult question. Move on and pick off the easy ones. Learning to recognize the questions you simply can't solve is a very, very important test-taking skill.

2) Always look as confident as possible. It helps if you act like you didn't even remember there was a test that day. It really irritates other test takers.

3) There are no shortcuts. Focus. Don't forget whatever it was you learned in the prerequisite classes that brought you to where you are today. For example, you'll not get the idea of limits if you don't know the difference between a term and a factor.

4) Don't worry about freaks who blast through the entire exam in half the allotted time and still get 100%. There is nothing you can do about those folks. If you think you know one, take a look at hint #2 and don't be intimidated.

5) Forget that you own a calculator and learn how to think more clearly and more logically. Don't let the magic electronic buttons replace your brain.

Okay, now go out there and see if you can beat 45%! Well, maybe you will want to shoot a little higher than that.

3. Originally Posted by TKHunny
TKHunny's off-the-wall test-taking hints...

1) Never, ever stew over a difficult question. Move on and pick off the easy ones. Learning to recognize the questions you simply can't solve is a very, very important test-taking skill.

2) Always look as confident as possible. It helps if you act like you didn't even remember there was a test that day. It really irritates other test takers.

3) There are no shortcuts. Focus. Don't forget whatever it was you learned in the prerequisite classes that brought you to where you are today. For example, you'll not get the idea of limits if you don't know the difference between a term and a factor.

4) Don't worry about freaks who blast through the entire exam in half the allotted time and still get 100%. There is nothing you can do about those folks. If you think you know one, take a look at hint #2 and don't be intimidated.

5) Forget that you own a calculator and learn how to think more clearly and more logically. Don't let the magic electronic buttons replace your brain.

Okay, now go out there and see if you can beat 45%! Well, maybe you will want to shoot a little higher than that.
Haha, those are funny. That is actually a good point though (#2). I definitely will do that.

Thanks!

4. Originally Posted by Coco87
(#2). I definitely will do that.
Oh, great. You picked the only one that doesn't actually help.

5. Originally Posted by Coco87
Haha, those are funny. That is actually a good point though (#2). I definitely will do that.

Thanks!
My suggestion would be to sometimes use the inverse of a composite function instead of chain rule.

Example: Find the derivative of

$\displaystyle y=\sqrt{\frac{x^2-5x+18}{x^3-17x^2+7}}$

Instead of doing chain rule just do this

$\displaystyle y^2=\frac{x^2-5x+18}{x^3-17x^2+7}$

so we get the derivative to be

$\displaystyle 2y\cdot{y'}=\frac{-(x^4-10x^3+139x^2-626x+35)}{(x^3-17x^2+7)^2}$

now solve for y' and remember that y is the original equation

6. Originally Posted by Coco87
Hey,
I took a test this Monday on Derivatives, and ran into a problem. I didn't have enough time to finish the test. We got the test back today, and I got all of my answer right, except I didn't get to the rest of the test, and ended up getting a 36%. Supposedly a few others are having the same problem, and he is letting the class re-take the test (different form though), except he is giving us 2 hours instead of 1 and a half. Now, I know how to find derivatives and have done plenty of homework. My problem is taking forever to complete them. Would anyone happen to have any tips or tricks at finding the derivative of a problem quickly? Even with the different rules for making the derivative easier to find, it still takes a while.

Thanks!
I skip the basics of differentiation, assuming that you know them.

The next thing you should master is here:

Learn to apply the chain rule without writing.

First, Memorize this derivative: $\displaystyle \frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$ ... Never forget it. It will save a lot time. No, don't think of it as $\displaystyle x^{\frac{1}{2}}$ and then calculate the derivative, just memorize it.

Now, chain rule:

$\displaystyle \frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$

Memorize this:
Derivative of f(g(x)) is f prime (g(x)) times g prime (x). Or shortly "derivative of f o g is (f prime g) times (g prime)."

Now you know the rule, time for learning to seperate functions into f and g without writing.

For example, when you see $\displaystyle \sqrt{\frac{x^2-1}{x-2}}$, you should directly separate this into $\displaystyle f(x) = \sqrt{x}$ and $\displaystyle g(x) = \frac{x^2-1}{x-2}$, and then apply the chain rule. NEVER use paper while doing this. NEVER. By using paper in this step, you can never calculate derivatives fast enough. After applying the chain rule, write your result down and simplify it. Don't forget to practice, practice and practice.

When you totally master this, you are the real derivative hero. You are the fastest.

I can't tell you how useful this technique is. It saves sooooo much time..

Example:

$\displaystyle y=\sin 3x$

$\displaystyle f(x) = \sin x$, $\displaystyle g(x)=3x$

$\displaystyle y'=\underbrace{\cos 3x}_{f'(g(x))} \cdot \underbrace{\left ( 3x \right )'}_{g'(x)}$

$\displaystyle y'=3 \cos 3x$

Example:

$\displaystyle y = a^{5x^3}$

If you can't see how to separate it, you should use $\displaystyle f(x) = a^x$ and $\displaystyle g(x)=5x^3$.

$\displaystyle y = a^{5x^3} \ln a \cdot \left ( 5x^3 \right )'$

Try it on Mathstud's example:

$\displaystyle y=\sqrt{\frac{x^2-5x+18}{x^3-17x^2+7}}$

You basically separated the function and saw that the derivative is,

$\displaystyle y'=\frac{1}{2\sqrt{\frac{x^2-5x+18}{x^3-17x^2+7}}}\cdot \left ( \frac{x^2-5x+18}{x^3-17x^2+7} \right )'$

$\displaystyle y=e^{\sin x}$

$\displaystyle y' = e^{\sin x}\cdot \left ( \sin x \right )'$

$\displaystyle y' = e^{\sin x}\cdot \cos x$

$\displaystyle y = \sin (\cos x)$

$\displaystyle y' = - \cos (\cos x)\cdot \sin x$

$\displaystyle y = \frac{1}{\cos x}$

$\displaystyle y' = -\frac{1}{\cos^2 x}\cdot \left ( \cos x \right )'$

$\displaystyle y' = \frac{\sin x}{\cos^2 x}$

There are countless of derivatives to find. You'll get faster and better as you practice. I hope this helps you. Good luck!

7. The others have done a pretty job covering things. Let me just mention a few more points to keep in mind:

-Unless you know your professor will be taking off a lot of points for it, I suggest you leave your answers mostly unsimplified at first. Then, when (and if) you have time, go back and simplify them. A bunch of unsimplified but correct answers should be worth more than a small selection of pretty and neatly cleaned up work. For example, after applying the quotient rule, just leave it in its big messy form and then go back later to expand and simplify.

-Don't erase unless you really need the space. You can spend a lot of time erasing things. I suggest you just cross things out when you make a mistake (but still try to keep things fairly neat, and make sure it is clear which of your work is crossed out and which isn't)

-Get the easy problems out of the way and save the longer ones for later. This ensures you don't miss 80% of the questions because you spent an hour trying to figure out some convoluted problem. It may help to just take 30 seconds or so at the beginning of the test to look through the problems so that you can see which ones are easy and which are difficult, rather than simply starting at number 1 and going in order.

-Try not to be too careful with calculations and such. While you certainly don't want to make an unnoticed error at the beginning of a long and difficult problem only to have to go back and fix 15 minutes' worth of work, you can be a little more carefree on the shorter problems, and you should hold off on checking most of your work until the end (and of course, if you ever finish early, always take the spare time to go back over your work).

-Read through your textbook outside of class and work through all of the examples (even if you are a busy person, you can usually find time for this during your "off-moments," e.g., in-between classes, waiting for the bus, etc). Do plenty of problems--practice is very important (this means that if you aren't comfortable with something do some additional problems even if you already did all of the homework).

-Make sure that you will be comfortable during the test. This includes a lot of things, but for example, wear comfortable clothes (especially if you know it will be very hot or very cold that day), use the restroom before the exam, turn off your cell phone to prevent distraction, etc. And when taking the exam, although you must work hard, you should also try to keep yourself fairly relaxed. Being comfortable will make it much easier for you to concentrate on your work.

-Finally, get plenty of rest, and eat a good breakfast (or lunch or dinner, depending on how late the class is!). This will keep you focused and your mind sharp.

8. Thank you everyone for your help! One of my issues was actually screwing around with the Chain rule too long. I only had 3 problems I didn't get to work on, but they were word problems, so it's not that bad. If I only got those 3 wrong, then I should come out with a 88% (so close to a 90%!!).

Thanks again!