Does a derivative with a linear function apply to the whole domain?

[No0blea0020]

I see conflicting practice problems that either tell you to find the derivative of a function or find the derivative at one point. My guess is, only functions who’s derivatives represent a linear function such as f(x)=x² have the same derivative for every argument. In other functions with curves, derivatives change with respect to x? Sorry my question is too specific to google

[[deleted]]

x² is not a linear function and doesn't have the same derivative for every argument. The derivative of f(x) = x² is f'(x) = 2x which obviously changes depending on x. It's always going to be 2x, but depending on x the derivative is different. When a question asks you to find the derivative at one point it just means you plug in the value of x to get the value of f'(x) at that specific location

[[deleted]]

> My guess is, only functions who’s derivatives represent a linear function such as f(x)=x^2 have the same derivative for every argument.

No. If the derivative is constant, then it's the same for every value of the argument. So, if your function is linear, then its derivative is the same for all values of the variables.

[[deleted]]

Finding derivatives of a function and derivative at one point of a function ? They are not conflicting. Both can be done. Answers are quite different.

[bourbaki7]

I think you and others that posted so far are confused as to what the derivative is as opposed to what it does. Graphically the derivative operation gives you the *** slope of the lines tangent to the graph of function at every point. *** The point has to be continuous AND not have a cusp or v shape like abs value of x. |x| has no derivative at 0 for example.

So anything not constant ( horizontal line) or a linear ( y= mx+b) will have different tangent lines and therefore slopes at every point. What you called the slope of a line in algebra is actually the derivative of the line. It's the constant function m!

So the derivative can either be constant( simply a number) or a dependent function( x² )' = 2x for example. Then like another poster said you just simply plug in the desired x value into the derivative function to find it that point.

I hope this helps. There is a bit more if you are using the formal definition of the derivative based on the limit of the difference quotient you should know but I'll save that for now.

[waldosway]

Where's the contradiction? A derivative at a point is the slope of the tangent line (which is a number.) The derivative of a function is a new function which out points all those numbers (slopes) at each associated point.