# Calculus Calculation Contest!

## Time yourself on this! What's the 4th derivative of x²sin(2x²)?

2 vote(s)
13.3%

4 vote(s)
26.7%

6 vote(s)
40.0%
4. ### I rather stick with IMO math olympiad questions.

3 vote(s)
20.0%
1. Mar 22, 2008

### JWHooper

I'll work on it, and I'll show my time.

2. Mar 22, 2008

### JWHooper

65 seconds.

3. Mar 22, 2008

### Crosson

I voted for 10 seconds. Using binomial coefficients this derivative is just a matter of writing the answer:

http://en.wikipedia.org/wiki/Leibniz_rule_(generalized_product_rule)

Admittedly, I didn't simplify it, in which case I would fall into 10-60 seconds.

4. Mar 23, 2008

### Gib Z

Is there a "I think it's a waste of time" option?

5. Mar 23, 2008

### ice109

is there a book somewhere of all these cute little tricks like the generalized product rule? and don't go telling me that i should be able to figure em out on my own.

6. Mar 23, 2008

### Crosson

I don't know about an entire book devoted only to tricks, but I first learned that one from Mary Boas' Mathematical Methods for the Physical Sciences. I think you will find physicists use these kinds of tricks most often; it has become common in the math department to say "here are the basic rules, specific examples are a waste of time."

Be careful, teaching introductory calculus is the main source of employment for math PhDs (from the perspective of university administration). Calling any part of math "a waste of time" is a slippery slope, since it quickly becomes hard to justify any of it as time well spent.

7. Mar 24, 2008

### Gib Z

Finding the fourth derivative of some product is really quite pointless though no? The vast majority of geometrical and physical applications require only up to the 2nd or 3rd derivative at most.

PS. It takes me 20 seconds if you let me leave the answer in series form. Don't ask me why I did it.

Last edited: Mar 24, 2008
8. Mar 24, 2008

### ice109

do you read the posts in the threads you post in? did you not see the link to the leibniz identity like 3 posts back?

9. Mar 24, 2008

### BrendanH

Does anyone dare give their answer?

10. Mar 25, 2008

### Gib Z

Do you read the links posted? That identity is in terms of lower derivatives, but derivatives still. What i meant was the write the sine term in its series form, multiply through by the x term and finding the 4th derivative of the resulting series.

11. Mar 25, 2008

### Crosson

The irony is that the direct way to calculate coefficients of a power series is by evaluating higher order derivatives! Of course this is almost never done in practice since we keep working with the same 10 functions who p-series we know by heart, but Liebniz's formula is very helpful once we venture out of the familiar functions, e.g.

Prove:

$$\sqrt{\frac{\pi }{2 x}} J_{\frac{1}{2} (2 n+1)}(x)=x^n \left(-\frac{x^{-1} d}{ dx}\right)^n \frac{\sin (x)}{x}$$

where the bessel function $J_p(x)$ is given by:

$$J_p(x)=\sum _{n=0}^{\infty } \frac{(-1)^n \left(\frac{x}{2}\right)^{2 n+p}}{\Gamma (n+1) \Gamma (n+p+1)}$$

If I remember correctly I spent ~2 hours working on this problem as a sophomore, and along the way I used a lot of tricks; Liebniz's rule was essential, and the most difficult part was getting the factorial terms to match up.

Last edited: Mar 25, 2008
12. Mar 25, 2008

### daveyinaz

As a sophomore? Man, you must have gone to a better school than I did.

13. Mar 25, 2008

### Crosson

Not really, I went to a fourth tier state school in my home town. Fortunately I had a good physics professor, and there was another good student at that time as well. That problem was given to us as part of a take-home test in the math methods course that used the Boas book I mentioned earlier. Most classmates showed that the first few terms of the two series were equal, and that received full credit! What a strange education I've had...