How Do You Find the Double Derivative of This Function?

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Discussion Overview

The discussion revolves around finding the second derivative of a function defined implicitly by the equation x^2 [f(x)]^4 + xf(x) = 6, given that f(2) = 1. Participants explore methods for differentiation, particularly focusing on implicit differentiation and the application of the product and chain rules.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the problem and requests a step-by-step solution.
  • Another participant suggests that the product rule and chain rule are necessary for solving the problem.
  • Some participants indicate that implicit differentiation is required, with one noting they are unsure how to proceed after finding the first derivative.
  • A participant provides a derived expression for the first derivative but questions its correctness and notes an extra term.
  • There is a proposed general plan for solving the problem, which includes differentiating, evaluating at x=2, and solving for f'(2) and f''(2).
  • One participant offers a tip regarding the differentiation of f(x)^4, emphasizing the use of the product rule.

Areas of Agreement / Disagreement

Participants generally agree that implicit differentiation is necessary and that the product and chain rules are applicable. However, there is no consensus on the correctness of the derived expressions or the best approach to take, leading to some confusion and uncertainty in the discussion.

Contextual Notes

Some participants express uncertainty about the steps involved in implicit differentiation and the application of differentiation rules, indicating potential gaps in understanding or missing assumptions about prior knowledge.

ozzie6616
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Okay so I have this calculus teacher who is crazy and gave us this problem on an exam. I don't know how to figure it out and I tried getting explanations but they were really hard and complicated.:confused: :confused: :confused: So here is the problem

x^2 [f(x)]^4 + xf(x) = 6

f(2) = 1 Find f''(2)

I would most greatly appreciate it if someone showed me how to solve the problem step by step.
:smile:
 
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Hi ozzie,
I hope you know how to differentiate using the product rule and chain rule, because those are the only things you need to solve the problem. Try that out and report any difficulty you might have.
 
I assume it is meant to differentiate the equation first one; from that determine f'(2), and then differentiate again and determine f''(2).
 
that looks like it requires some implicit differentiation, which i haven't encountered since calc I, so i don't really know how to do that one.
 
neutrino said:
Hi ozzie,
I hope you know how to differentiate using the product rule and chain rule, because those are the only things you need to solve the problem. Try that out and report any difficulty you might have.
Is one method preferable to the other?
 
dimensionless said:
Is one method preferable to the other?
Both are required.
 
yes it does require implicit differentiation. I have gone through the first step to find the first derivative, however when I get stumped when finding the second derivative.

This is how far i got and I don't know if its right.
2x[f(x)]^4 + 4[F(x)]^3[f'(x)]x^2 + f(x) + f(x) + f'(x)x=0

Now I don't know what to do next.
 
ozzie6616 said:
yes it does require implicit differentiation. I have gone through the first step to find the first derivative, however when I get stumped when finding the second derivative.

This is how far i got and I don't know if its right.
2x[f(x)]^4 + 4[F(x)]^3[f'(x)]x^2 + f(x) + f(x) + f'(x)x=0

Now I don't know what to do next.

There is an extra f(x). Now substitute x = 2 and f(2) = 1 to find f'(x). And differntiate the above equation once more to get one the expression for f''(x). Plug-in the known values at x=2 and find f''.
 
so, the general plan is:
-differentiate once with respect to x
-evaluate the expression @x=2
-solve for f '(2)
-differentiate again with respect to x
-evaluate the new expression @x=2
-solve for the only unknown f ''(2)

a tip:

d/dx[ f(x)^4 ] = 4*f '(x)*f(x)^3

so the first derivative should be:

x^2*4*f '(x)*f(x)^3+2x*f(x)^4+x*f '(x)+f(x)=0

that is using the product rule on both terms of the original equation.

to further understand the differentiation of a general function f(x)
take careful note that: d/dx[f(x)]=f '(x)*d/dx(x)
so for example d/dx[ f(x^2) ]= f '(x^2)*2x
 

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