Find f'(x): Product Rule for Derivative of (3x^4)(6/(3√x^5)) (6x^2 - 3x)^5

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Homework Help Overview

The discussion revolves around finding the derivative of a function expressed as a product of multiple terms, specifically f(x) = (3x^4) [6/(3√x^5)] (6x^2 - 3x)^5. The context includes implicit differentiation and simplification of expressions involving variables a and b, which are later clarified to represent x and y.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the application of the product rule for derivatives and the use of implicit differentiation. There are questions about the clarity of the original problem statement and the appropriateness of the methods being used. Some participants express confusion regarding the notation and the implications of the equal sign in the context of implicit differentiation.

Discussion Status

There is an ongoing exploration of the methods to differentiate the given expressions. Some participants have offered guidance on the use of the product rule and implicit differentiation, while others are seeking clarification on the original problem's intent and structure. Multiple interpretations of the problem are being discussed, indicating a productive exchange of ideas.

Contextual Notes

Participants note the potential confusion arising from the notation used in the problem statement, particularly regarding the variables a and b, which were initially misidentified. There is also mention of the need to set equations equal to zero for implicit differentiation, highlighting the constraints of the problem setup.

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Homework Statement

Find dy/dx: 4a^2b - 3b^2 = a^3 Please SIMPLIFY this one

Find f'(x): f(x) = (3x^4) [6/(3√x^5)] (6x^2 - 3x)^5 The middle is 6 divided by the cube root of x^5, Please do not simplify fully for this one and use the product rule

The Attempt at a Solution



i got - (8ab - 3a^2)/(4a^2 - 6b) using the implicit rule
However I need some help simplifyingFor the second one, would my approach be (derivative of first)(original second)(original third) + (derivative of second)(original first)(original third) + (derivative of third)(original first)(original second) ?
 
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Think makes no sense to me.

What are you trying to do?
 
Your approach for the second is fine. The first is unintelligible. Please check what you wrote.
 
haruspex said:
Your approach for the second is fine. The first is unintelligible. Please check what you wrote.

Is this not the result of the implicit method? Should I not even be using the implicit method? I thought I had to with numerator being the derivative with respect to "a" and the denominator the derivative with respect to "b"
 
Stanc said:
Is this not the result of the implicit method? Should I not even be using the implicit method? I thought I had to with numerator being the derivative with respect to "a" and the denominator the derivative with respect to "b"

How should we know what you need to do if its not even clear what you're question asking?

You just simply said Find dy/dx, then gave an expression in terms of a and b then said simplify.

What does that even mean?
 
Jorriss said:
How should we know what you need to do if its not even clear what you're question asking?

You just simply said Find dy/dx, then gave an expression in terms of a and b then said simplify.

What does that even mean?

Sorry about the a and b, it should be a=x and b = y (teacher gave it to us like that) but the question is to simply find the derivative (dy/dx)... i don't know what more I can tell you...
 
Stanc said:
Sorry about the a and b, it should be a=x and b = y (teacher gave it to us like that) but the question is to simply find the derivative (dy/dx)... i don't know what more I can tell you...

Ok, but it still doesn't make sense: Find dy/dx: 4x2y - 3y2 = x3. What's the equals sign doing?
 
haruspex said:
Ok, but it still doesn't make sense: Find dy/dx: 4x2y - 3y2 = x3. What's the equals sign doing?

I have to set it equal to zero and use implicit differentiation
 
Stanc said:
I have to set it equal to zero and use implicit differentiation
Sorry, I was confusing two questions. Your answer looks right.
 
  • #10
haruspex said:
Sorry, I was confusing two questions. Your answer looks right.

Ok Thank you! but is that all it can be simplified down to?
 
  • #11
Stanc said:
Ok Thank you! but is that all it can be simplified down to?
You could factorise out an x in the numerator, but other than that, that's as simple as it will go.
 
  • #12
haruspex said:
You could factorise out an x in the numerator, but other than that, that's as simple as it will go.

May I ask you one more question? How would i find the derivative of (1/x) + (1//y) = 1 using implicit and for a question like (x^2/19) - (y^2/9) = 1 would I find the common denominator and then go through with implicit differentiation?
 
  • #13
Stanc said:
May I ask you one more question? How would i find the derivative of (1/x) + (1//y) = 1 using implicit and for a question like (x^2/19) - (y^2/9) = 1 would I find the common denominator and then go through with implicit differentiation?

You can just differentiate straight away. What's the difficulty? Yes, multiplying out first might make it a little simpler.
 

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