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anil86
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MHB Find nth Derivative - Quick Tutorial
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- Derivative Tutorial
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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
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Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.
- #3
anil86
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Opalg
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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.
Hello!
I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem.
Given:
##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0##
##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1##
##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0##
I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
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