Prove Power Rule: Implicit Differentiation w/ Rational Exponents

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SUMMARY

The discussion focuses on proving the power rule for differentiation, specifically for rational exponents, expressed as d/dx[x^(p/q)] = (p/q)x^((p/q)-1). The method involves rewriting the equation as y^q = x^p and applying implicit differentiation. Participants confirm the assumption that the power rule is valid for integer exponents, which simplifies the differentiation process. The conversation emphasizes the importance of correctly applying implicit differentiation to derive the result.

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  • Understanding of implicit differentiation
  • Knowledge of rational exponents
  • Familiarity with the power rule of differentiation
  • Basic algebraic manipulation skills
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  • Practice differentiating functions using the power rule
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Students learning calculus, educators teaching differentiation techniques, and anyone seeking to deepen their understanding of implicit differentiation and rational exponents.

mattxr250
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Ok guys, I'm new here and I need some help with a math problem...

The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...

the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p and differentiate implicitly...assume that p and q are integers, where q>0.

Thanks for any help

Matt
 
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Would I be correct in guessing that we are allowed to assume that the power rule holds for integers? If so, then it is simply a matter of differentiating both sides of the equation [tex]y^q=x^{p}[/tex] and then solving for y'. Where are you stuck?
 
Don't multiple post.
 

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