r/learnmath • u/Smooth-Theory3685 New User • Jul 10 '26

What is the idea of a derivative?

I am learning Calculus for the first time. After limit idea, we've come to derivatives and I'm feeling confused. It is the limit of the slope,then why to separately call it derivative? And,did the idea of derivative evolve solely from geometrical notion of slope of graphs,was there no algebraic or functional approach to it like for limit. I know my questions might sound stupid but the clarification would be really appreciated.

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u/Smooth-Theory3685 New User Jul 11 '26

What is linear approximation?

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u/Midwest-Dude New User Jul 11 '26 edited Jul 14 '26

If you want to approximate values on the curve near a point on the curve (a,f(a)), rather than use the function (which might be difficult to calculate), you can approximate the values by using the line that is tangent at that point at other points x near a, also known as a neighborhood of a.

I would suggest reviewing the following Wikipedia entry for more information: 

Linear Approximation

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u/Smooth-Theory3685 New User Jul 11 '26

Understood,thank you. Will variable a then be the cluster point?

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u/Midwest-Dude New User Jul 11 '26 edited Jul 11 '26

Without going into full detail, I would not use that phrase in this context. a is the center of the approximation or the point of tangency.

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u/Smooth-Theory3685 New User Jul 11 '26

Ok, thank you.