I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring point ##x=x_{0}+\Delta x##.(adsbygoogle = window.adsbygoogle || []).push({});

What confuses me is how to interpret its meaning correctly, that is, what does the derivative ##f'(x_{0})## actually describe?

On Wikipedia it says that"the derivative of a function ##f(x)## of a variable ##x## is a measure of the rate at which the value of the function changes with respect to the change of the variable". However, the function has a particularvalue, ##f(x_{0})## at a given point ##x=x_{0}## so how can one meaningfully discuss the rate at which the value of the function is changing at that point?constant

Would it be correct to interpret the derivative of a function at a point as quantifying the slope of the line tangent to the curve (corresponding to the coordinate plot of the function ##f(x)##) at that point? Is the whole idea that when taking the derivative of a function one isn't ever considering the value of a function at a given point, one considers the function over a given neighbourhood. Upon taking the derivative one subsequently evaluates it at a given point and this quantifies the instantaneous rate of change in the value of the function (with respect to a change in its input variable ##x##) at that point. For a linear function, this rate of change is constant, i.e. the rate at which the value of the function is changing (with respect to a change in its input variable ##x##) at each point is the same. For a more general (continuous) function, the derivative will itself be a function and its value will change from point to point, however, at each given point its value at that point will still equal the slope of the line (corresponding to a the coordinate plot of a linear function) tangent to the curve (corresponding to the coordinate plot of the more general function we're considering) at that point?!

Apologies for such a basic question, but I've really got a mental block in my head about this and want to clear it up.

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# B What does the derivative of a function at a point describe?

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