How do you solve for x when using the chain rule and functions with variables?

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Discussion Overview

The discussion revolves around differentiating the function y = (4-x) ^ 5x using the chain rule and product rule in calculus. Participants explore various methods for finding the derivative, including implicit differentiation and logarithmic differentiation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant initially presents the function and seeks guidance on how to differentiate it.
  • Another participant suggests that the question may relate to finding a differential equation and proposes taking logarithms to differentiate.
  • A participant clarifies that the task is differentiation and acknowledges the need for further instructions.
  • Instructions are provided to take logs and differentiate implicitly, leading to a proposed form for dy/dx.
  • A participant shares their derived expression for dy/dx but is cautioned about the product rule and the need to include the natural logarithm in their calculations.
  • Further clarification is offered regarding the differentiation process, emphasizing the importance of recognizing the product rule in the expression.
  • Another participant introduces an alternative approach using partial derivatives and the chain rule, defining u and v as functions of x.
  • One participant reflects on their understanding and acknowledges the importance of practice in mastering these concepts.
  • Another participant provides a general formula for differentiating functions of the form T^u, suggesting to substitute specific values for t and u.

Areas of Agreement / Disagreement

Participants generally agree on the need to differentiate the function but present multiple approaches and methods, indicating that no single consensus exists on the best method to apply.

Contextual Notes

Some participants express uncertainty about specific steps in the differentiation process and the application of rules like the product rule and chain rule. There are also mentions of missing assumptions and the complexity of the function involved.

Who May Find This Useful

Students and individuals interested in calculus, particularly those looking to understand differentiation techniques involving functions of variables and the application of the chain and product rules.

cogs24
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y = (4-x) ^ 5x

Just wondering what i should do with this
Thanx
 
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I guess that depends what the question is. Are you asking for the differential equation to which this is the solution?

Take logs and differentiate for first order ODE?
 
yes, i have to differentiate it

sorry i didnt specifiy what had to be done, i presumed you knew it was differentiation.
 
If it's just finding the derivative then this is probably more a question for the calculus thread.

Anyhoo, i'll do the first part. Take logs to get.

ln(y)= 5x ln(4-x)

Now differentiate (implicitly on the LHS) and rearrange to get your answer for dy/dx. Have a go and let me know if/where you get stuck.
 
ok, i understand your instructions, its just a matter of doing the right things now
This is what i got as an answer, unfortunately we arent supplied with answers in this exercise.

following on from your step, this is what i did

1/y * dy/dx = 5 * (1/4-x)
dy/dx = (5/4-x)y
dy/dx = (5/4-x)(4-x)^5x
 
Pay attention with the differentiaition of the RHS.It's a product.I'm sure one of the 2 terms will contain the natural logarithm.

Daniel.
 
Think with my poor Tex you missed the x after the 5 on the RHS. Take that into account and you're there.
 
cogs24 said:
y = (4-x) ^ 5x

Just wondering what i should do with this
Thanx
Here's another way:
Let y(x)=f(u(x),v(x)), f(u,v)=u^{v}, u(x)=4-x, v(x)=5x
Then, we have:
\frac{dy}{dx}=\frac{\partial{f}}{\partial{u}}\frac{du}{dx}+\frac{\partial{f}}{\partial{v}}\frac{dv}{dx}
 
ahh i see. i didnt spot the product rule on the right, i guess practice makes perfect.
Thanx everyone for the input.
 
  • #10
when differentiating functions of a variable say x raised to another function of xthen let's assume they are T^u,where t and u are functions of x,dy/dx =
T^u[du/dx*logt+u(dt/dx)/t]
 
  • #11
when differentiating functions of a variable say x raised to another function of xthen let's assume they are T^u,where t and u are functions of x,dy/dx =
T^u[du/dx*logt+u(dt/dx)/t].now let 4-x=t,and u=5x.
take normal procedures and see if it works
 

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