The derivative is the slope of a curve at a point, but that phrase hides how it is actually built. Take two points on a curve and join them — that is a secant, and its slope is easy to find. Now slide one point toward the other. The secant turns into a tangent, and that slope is the derivative. Here you draw your own curve and watch it happen.
Draw a curve with your finger or the mouse — anything you like. Three tabs then appear. In Secant and Tangent, drag the two points A and B. In Local Linearity, use the zoom slider or press Play to zoom in automatically. In Sign Analysis, drag the single point P along the curve. Clear erases everything and starts over.
In the first tab, the line through A and B with its slope shown. Bring B close enough to A and it becomes the tangent. In the second, zooming in far enough makes any smooth curve look like a straight line — that is what differentiable means. In the third, the curve is coloured by the sign of its derivative: rising, falling, or flat at a peak or trough.
Your curve is smoothed slightly as you draw it. Slopes are pure numbers with no units. The derivative is worked out numerically, not from a formula.
You should be able to say why zooming in on a smooth curve always gives you a straight line, and why that is what makes a derivative possible.