The power rule and the chains rule

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SUMMARY

The discussion clarifies the application of the power rule and the chain rule in calculus. The power rule is applicable when differentiating functions of the form x^n, where n is a constant, such as x^2 or 3x^6. Conversely, the chain rule is employed when differentiating composite functions that include variables in the exponent, exemplified by the differentiation of e^(2x), which results in 2e^(2x) due to the inner function 2x. Understanding these distinctions is crucial for accurate differentiation.

PREREQUISITES
  • Understanding of basic calculus concepts, including differentiation.
  • Familiarity with the power rule for derivatives.
  • Knowledge of the chain rule for composite functions.
  • Ability to identify functions and their components in calculus problems.
NEXT STEPS
  • Study the application of the power rule in various polynomial functions.
  • Explore more complex examples of the chain rule with nested functions.
  • Practice problems that require the simultaneous use of both the power rule and the chain rule.
  • Review the implications of the product and quotient rules in conjunction with the power and chain rules.
USEFUL FOR

Students studying calculus, educators teaching differentiation techniques, and anyone seeking to master the application of the power and chain rules in mathematical analysis.

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Homework Statement



How do I know for sure when to use the power rule instead of the chain rule and vice versa?

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The Attempt at a Solution

 
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power rule is used when there are no variables in the raised part (ie. x^2 or 3x^6) but chain rule is used if there are variables in the raised part (ie. d/dx (e^(2x))=e^(2x) * d/dx (2x) = 2e^(2x) Chain rule is used in other situations but this is basically the situation where you would have to decide between chain and power rule as you are asking.
 

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