Imagine you have a curve and you find the derivative of that curve.

It is a known fact that if the derivative is positive in certain areas then the curve is increasing in those areas. But why is this?

The derivative shows the gradient of a tangent at a certain point. So if you find the derivative to be positive for a certain region (say x=0 to x=2), you know that all the gradients of tangents are positive in that area. But people seem to assume the curve is increasing in this area as well why is this?

Sorry if i have worded the question badly it is hard to put into words, but i was just solving questions where is asked me to find where the curve was increasing and i started to think why positive gradients-of-tangents proved that the curve was increasing

thanks