Understanding the Chain Rule: A Derivative Problem Solution

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SUMMARY

The discussion centers on the application of the chain rule and product rule in calculating the derivative of the function f(x) = x²(x-2)⁴. The user initially applied the product rule incorrectly, leading to confusion regarding the simplification process. The correct derivative, as confirmed by another participant, is f' = 2x(x-2)³(3x-2), which matches the book's answer. The key takeaway is the importance of recognizing when to apply the chain rule within the context of product rule derivatives.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically derivatives.
  • Familiarity with the product rule and chain rule in differentiation.
  • Ability to simplify algebraic expressions involving polynomials.
  • Knowledge of function notation and manipulation.
NEXT STEPS
  • Study the application of the chain rule in more complex functions.
  • Practice problems involving both the product rule and chain rule for derivatives.
  • Learn techniques for simplifying polynomial expressions after differentiation.
  • Explore additional resources on derivative rules in calculus textbooks or online platforms.
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to improve their understanding of derivative rules and simplification techniques.

Phyzwizz
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I did a derivative problem, but my book says that my answer is wrong.

f(x)=x2(x-2)4

I didn't see much use in the chain rule so I used the product rule.

x2(4(x-2)3) + (x-2)4(2x)
=4x2(x-2)3 + 2x(x-2)4

The book says that instead of this, the answer is ...
x2(4(x-2)3(1)) + (x-2)4(2x) = 2x(x-2)3(3x-2)

So I guess where I'm really confused is in the manner with which the book simplified.
 
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Phyzwizz said:
I did a derivative problem, but my book says that my answer is wrong.

f(x)=x2(x-2)4

I didn't see much use in the chain rule so I used the product rule.

x2(4(x-2)3) + (x-2)4(2x)
=4x2(x-2)3 + 2x(x-2)4
Well, you did use the chain rule when you said that the derivative of (x- 2)4 is 4(x- 2)3 (because the derivative of x- 2 is 1).

If you factor out 2x(x- 2)3 to get f'= 2x(x- 2)3(2x+ (x- 2))= 2x(x- 2)(3x- 2), exactly what your book says. Your answer and your book's answer are the same.

The book says that instead of this, the answer is ...
x2(4(x-2)3(1)) + (x-2)4(2x) = 2x(x-2)3(3x-2)

So I guess where I'm really confused is in the manner with which the book simplified.
 
Awesome thanks, I get it now. And I also now understand how ridiculous my book is in its mindless factoring.
 

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