Is the Derivative of a Linear Function the Same as the Function Itself?

[rjvsngh]

Given that the derivative of a linear function is the function itself, how do I make sense of the following:
Given f(x) = x. It's derivative is g(x) = f'(x) = 1. Is g(x) the same as f(x) in some way? Or have I got this wrong in some way. Is f'(x) really the derivative of a f(x) in the sense of the statement that "the derivative of a linear function is the function"?

 

[HallsofIvy]

Where were you "given that the derivative of a linear function is the function itself"? The derivative of an exponential function is the function itself, not a linear function.

 

[Bacle]

I have seen this confusing treatment too, rjvsngh, and this is the best sense I have been able to make of it : I think they are conflating the terms
derivative with differential. The differential is the linear map that best approximates
(locally) the change of a (differentiable function) , so that, e.g., a line with slope
2x is the best local linear approximation to the change of f(x)=x^2 , so, by derivative,
they mean differential, and the differential (at xo) is then is y-yo=2xo(x-xo). But then,
if your function is (globally) linear to start with, then the best linear approximation, aka,
differential , is the function itself. So, the differential of a linear function L is L itself,
but the derivative of L itself is not L.

 

[1mmorta1]

The derivative of a linear function is the slope of the function, m. It is a constant, rather than a function. i.e. f(x) = mx + k, f'(x) = m. So in this case, g(x) isn't a function at all, but a number.
f'(x) = g(x) is a differential equation, and therefor a whole different animal.

 

[lavinia]

Given that the derivative of a linear function is the function itself, how do I make sense of the following:
Given f(x) = x. It's derivative is g(x) = f'(x) = 1. Is g(x) the same as f(x) in some way? Or have I got this wrong in some way. Is f'(x) really the derivative of a f(x) in the sense of the statement that "the derivative of a linear function is the function"?

f(x) = x = 1.x

You need to view the derivative as a mapping.

 

[Bacle]

But this is not the standard difference between derivative as an operator or as an
element of the dual; it is an assignment of the differential , not the derivative.

 

[lavinia]

But this is not the standard difference between derivative as an operator or as an
element of the dual; it is an assignment of the differential , not the derivative.

Right, correct language would call it the differential.

 

[Bacle]

Right, Lavinia, I wish I had known when I first ran into this layout.

 

[rjvsngh]

thanks for all these explanations. following a particular reply, i did realize my question was incorrect in the usage of terms. my question originated in something i read in "Math Analysis", Apostol, 2nd ed., in the chapter on multi-variable calculus. However, looking closely, the precise statement was that "the total derivative of a linear function is the function itself" and i now realize that the total derivative as defined by Apostol and the simple derivative are different. in fact, Apostol does point this out in the text as well.

i guess the "derivative as a number" notion arises in the serendipitous (?) fact that a linear functional on R¹ amounts to multiplication by a number - the number being the so-called derivative.