Let AB be the secant line, passing through the points and . If , that is, approaches zero, then the secant line approaches the tangent line at the point . Accordingly, the slope of the tangent line is the limit of the slope of the secant line when approach zero:

Thus, the derivative can be interpreted as the slope of the tangent line at the point on the graph of the function .

If a function is increasing on some interval, then the slope of the tangent is positive at each point of that interval, and hence, the derivative of the function is positive.

If a function is decreasing on some interval, then the slope of the tangent is negative at each point of that interval, and hence, the derivative of the function is negative.

If a function is increasing on some interval, then the slope of the tangent is positive at each point of that interval, and hence, the derivative of the function is positive.

If a curve y = f (x) has a smooth top then the peak of the curve serves as the boundary between intervals of increasing and decreasing of the function. At this point the tangent is parallel to the x-axis. Therefore, its slope equals zero, and so .

Likewise, if a curve y = f (x) has a smooth bottom then there exists a point peak of the curve serves as the boundary between intervals of increasing and decreasing of the function. At this point the tangent is parallel to the x-axis. Therefore, its slope equals zero, and so .