Define derivative from 3 perspectives?

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SUMMARY

The discussion defines the derivative from three perspectives: as the slope of a tangent line, as the rate of change, and as the reverse of integration for continuous functions. Participants clarify that while differentiation is related to the derivative, they are not synonymous. The mention of Der(C^(oo)(R)) introduces a more advanced mathematical concept, emphasizing the derivative's role in higher-level calculus.

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  • Understanding of basic calculus concepts, including limits and continuity.
  • Familiarity with differentiation and its applications in mathematics.
  • Knowledge of integration and its relationship to differentiation.
  • Basic grasp of advanced calculus terminology, such as Der(C^(oo)(R)).
NEXT STEPS
  • Study the concept of limits in calculus to strengthen foundational knowledge.
  • Explore the relationship between differentiation and integration in calculus.
  • Learn about the geometric interpretation of derivatives, specifically the slope of tangent lines.
  • Investigate advanced calculus topics, including the implications of Der(C^(oo)(R)) in functional analysis.
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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of derivatives and their applications in various mathematical contexts.

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Define derivative from 3 perspectives?

I might be off but I only came up with slope and rate of change?
Is this correct? Whats the 3rd one?

Thanks for any help in advance !
 
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Slope of the tangent line.

(Which is just a different way of saying what matt grime said.)
 
How about "the reverse of integration (on continuous functions)"

Or an element of Der(C^(oo)(R)) if you want to be fancy.

OK so that's what differentiation is, not exactly the same as the derivative.
 

Thread 'Ambiguity of the term "indefinite integral"'

I've encountered a few different definitions of "indefinite integral," denoted ##\int f(x) \, dx##. any particular antiderivative ##F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)## the set of all antiderivatives ##\{F:\mathbb{R} \to \mathbb{R}, F'(x) = f(x)\}## a "canonical" antiderivative any expression of the form ##\int_a^x f(x) \, dx##, where ##a## is in the domain of ##f## and ##f## is continuous Sometimes, it becomes a little unclear which definition an author really has in mind, though...
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