- #1
anil86
- 10
- 0
MHB Find nth Derivative - Quick Tutorial
- Thread starter anil86
- Start date
-
- Tags
- Derivative Tutorial
Click For Summary
The discussion focuses on finding the nth derivative, but the main task is to prove a specific equation involving derivatives. The user is asked to re-scan an attachment that is currently unclear. The assignment clarifies that the goal is not to find the nth derivative but to demonstrate a relationship between derivatives. A base case is established using a previously derived equation, which can then be used for proof by induction. The conversation emphasizes the importance of clarity in shared materials and the correct interpretation of assignment requirements.
- #2
Ackbach
Gold Member
MHB
- 4,148
- 94
Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.
- #3
anil86
- 10
- 0
Attachments
- #4
Opalg
Gold Member
MHB
- 2,778
- 13
The assignment does not ask you to find the $n$th derivative, but to prove that $$(1-x^2)y_{n+1} - 2(\gamma + nx) y_n -n(n-1)y_{n-1} = 0.$$
You have shown that $$y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2}.$$ Write that as $$(1-x^2)y_1 - 2\gamma y = 0.$$ Differentiate, to get $$(1-x^2)y_2 - 2xy_1 - 2\gamma y_1 = 0.$$ Now use that as the base case for a proof by induction.
You have shown that $$y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2}.$$ Write that as $$(1-x^2)y_1 - 2\gamma y = 0.$$ Differentiate, to get $$(1-x^2)y_2 - 2xy_1 - 2\gamma y_1 = 0.$$ Now use that as the base case for a proof by induction.