derivative question confusion

Question

if we have a function f where $f(x)=x^2$ then the derivative function $f'$ can be calculated $f'(x)= 2x$, and this gives the derivative of f wrt x at some point x in the domain of f.

my confusion firstly is about the word derivative. Is it a number like "slope of a line" which may exist at every point x in a function and thus we can find the derivative function given a function.

Secondly if a function is defined by $y=x^2$ then i think $dy/dx$ measures the derivative at a point in this function and as more than one derivative exists we can express $dy/dx$ as a function so $dy/dx=2x$.

im really quite confused about this sorry if its badly worded but could you help me out here with my confusion. THanks

Answer 1

For each point $x$ of the domain $D_f$ of a function $f$, $f$ may have a derivative at $x$. So, yes, the derivative of $f$ is a function: it is the function which maps each $x\in D_f$ at which $f$ is differentiable to the derivative of $f$ at $x$.

This function is denoted by $f'$ or by $\dfrac{\mathrm df}{\mathrm dx}$. They mean the same thing.

Answer 2

The derivative can be understood as the instantaneous slope of a curve.

If you consider two close points on a smooth curve, you can approximate the curve with a line, and find the slope of this line. If you let the second point come closer and closer to the first, the line will converge to the tangent at the first point.

When you move the first point, that instantaneous slope can vary.