What Did I Do Wrong? Analyzing y = x tan x

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Discussion Overview

The discussion revolves around the second derivative of the function y = x tan x, specifically addressing a participant's confusion regarding their calculations and the discrepancies with a textbook answer. The scope includes mathematical reasoning and technical explanation related to differentiation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents their calculation of the second derivative, y'' = 2sec^3 x + sec^4(x), and expresses confusion about the discrepancy with the textbook result.
  • Another participant suggests that the first term in the differentiation should involve multiplying by tan x, leading to a correction in the derivative calculation.
  • A third participant points out that the initial calculation involved a multiplication error, indicating that the first "sec^2 x" was incorrectly multiplied instead of added.
  • A later reply proposes a different expression for the second derivative, suggesting it is 2sec^2 x(tan x + 1), but does not clarify how this relates to the previous calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct form of the second derivative, as multiple competing views and corrections are presented without resolution.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the differentiation process, particularly concerning the application of the product rule and the handling of secant functions.

PrudensOptimus
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y = x tan x

y' = tan x + x*sec^2(x)

y'' = sec^2(x) + d/dx (x*sec^2(x))

= ... + (d/dx(sec^2(x)) + sec^2(x))

= ... + (2secx + sec^2(x))

= 2sec^3 x + sec^4(x)

what did I do wrong?? in book it says y'' = something else.
 
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= ... + (2secx + sec^2(x))
First term should have d/dx(secx)=tanx.secx multipled, i.e. first term is 2sec2x.tanx.
 
= ... + (d/dx(sec^2(x)) + sec^2(x))

= ... + (2secx + sec^2(x))

= 2sec^3 x + sec^4(x)

You multiplied the first "sec^2 x" instead of adding!
 
I found the answer this morning,

something like

2sec^2 x(tanx + 1)
 

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