The idea of the derivative of a function
First we can tell what the idea of a derivative is. But the issue of computing derivatives is another thing entirely: a person can understand the idea without being able to effectively compute, and vice-versa.
Suppose that $f$ is a function of interest for some reason. We can give $f$ some sort of ‘geometric life’ by thinking about the set of points $(x,y)$ so that $$f(x)=y$$ We would say that this describes a curve in the $(x,y)$-plane. (And sometimes we think of $x$ as ‘moving’ from left to right, imparting further intuitive or physical content to the story).
For some particular number $x_o$, let $y_o$ be the value $f(x_o)$ obtained as output by plugging $x_o$ into $f$ as input. Then the point $(x_o,y_o)$ is a point on our curve. The tangent line to the curve at the point $(x_o,y_o)$ is a line passing through $(x_o,y_o)$ and ‘flat against’ the curve. (As opposed to crossing it at some definite angle).
The idea of the derivative $f'(x_o)$ is that it is the slope of the tangent line at $x_o$ to the curve. But this isn't the way to compute these things...