Is a function a derivative? I'm reading introductory calculus and I find that 'function' tends to be defined by what it does rather than what it is. If $y = f(x)$, then surely the value of $y$ is dependent on that of $x$, i.e. derived from that of $x$; hence $y$ is a derivative and thus 'function of' equals 'derivative of'. 
And yet the following chapters appear to deal with "derivative of a function", as if the word "derivative" is reserved for a particular type or phase of "derivation".
Can someone please clarify this for me? 
 A: A derivative is a function that is derived from another function, but take care not to miss out on the conceptual importance of the derivative as a rate of change as opposed to a simple derivation from another function.

Your issue seems to be primarily notational and linguistic rather than conceptual. 


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*When you write "$y=f(x)$," what do you really mean? You almost never see something like $y=f(x)=x^2$; that is, $y$ and $f(x)$ denote the same thing and using them together like that would certainly be considered redundant. 

*Have you ever heard of a mapping by any chance? The words function and mapping are synonymous, but you will frequently encounter the latter word later down the road and for good reason: the mapping given by $f(x)=x^2$, for example, assigns to each real number $x$ the real number $x^2$. This is often communicated by writing something like, "Let the mapping $\alpha\colon\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^2$." Hence, $\alpha$ is the name of the mapping that is defined by $f(x)=x^2$, where input values $x$ are turned into or "mapped to" the value $x^2$. This may sometimes be communicated by writing $x\mapsto x^2$ to indicate that $x^2$ is the image of $x$ under a mapping (in this case $\alpha$). 

*Your next issue seems to be with the linguistic interpretation of derive as opposed to derivative. 



Here is how the Oxford English Dictionary defines derivative in the mathematical sense:

Hence, it would be more appropriate to say that a derivative is a function derived from another, but you need to make sure you are not missing the point of what a derivative actually is conceptually: the rate of change of a function, often expressed as a function itself (though you may consider the local rate of change rather than a moral global rate of change; even so, you could still use a function to describe this). 
In this sense, the function $f'(x)=2x$ is the derivative of $f(x)=x^2$. Sure, $f'(x)$ is most certainly derived from $f(x)$, but you need to make sure you understand that what $f'(x)$ describes is the rate of change of $f(x)$ in a general sense. 
Does that clear things up? 
