In what sense the derivative is linear?. In Bartle's book The elements of real analysis takes as an observation the case f:R$\rightarrow$ R and says f is diferentiable at c iff the derivative f'(c) of f exists at c. In this case the derivative of f at c is linear function on R to R wich sends the real number u into the real number f'(c)u.
Then I took f(x)=$x^3$ , f'(c)=3$c^2$ at c$\neq$0 how do I define the derivative to be linear?. Obviously f'(a+b)$\neq$f'(a)+f'(b) for a, b $\neq$ 0 in general, so this isn't the sense, then I took $\phi$(x)=f'(c)(x) and yes this is linear for any f on R to R with derivative at c, because f'(c) is a number and $\phi$(x) is just a linear function in the sense $\phi$(x)=ax , where a is constant
So my question is this last sense the one to take to understand that a derivative at a point is a linear function? thanks beforehand.