If a function is increasing, what can be said about its derivative?

When a function is increasing, it means that as you move from left to right along the graph, the function values (outputs) steadily rise. This behavior directly relates to the concept of the derivative, which provides information about the rate of change of the function.

The derivative at any given point on the function tells us how steep the function is at that point. If the function is increasing, the value of the derivative must be greater than zero at every point in the interval where the function is increasing. A positive derivative indicates that for a small increase in the input (x-value), the output (y-value) increases, aligning perfectly with the definition of an increasing function.

This connection is the reason why when one observes an increasing function, its derivative is positively valued. In contexts where a derivative is exactly zero, the function is neither increasing nor decreasing, indicating a flat tangent line rather than a rise. If a derivative is negative, the function is decreasing. Lastly, if the derivative is undefined at a point, it typically suggests a discontinuity or a sharp corner in the graph, rather than a consistent increase.

Thus, the presence of a positive derivative is a clear and consistent identifier of an increasing function.