Why the Derivative of an Increasing Function is Positive: Understanding the Relationship between f(x) and f'(x)

If f(x) is increasing, then f'(x) is?

If f(x) is an increasing function, then its derivative, f'(x), will be positive

If f(x) is an increasing function, then its derivative, f'(x), will be positive.

To understand why this is the case, let’s first define what it means for a function to be increasing.

A function f(x) is said to be increasing on an interval if for any two points a and b in that interval, where a < b, the value of f(a) is less than f(b). In other words, if you move from left to right along the x-axis within the interval, the values of the function increase. Now, let's consider the derivative of f(x), which represents the rate of change of the function at a given point. If f(x) is increasing, it means that as x increases, so does the value of f(x). Geometrically, this means that the graph of f(x) slopes upwards as you move from left to right. The derivative, f'(x), gives us information about the slope of the graph of f(x). By definition, f'(x) represents the instantaneous rate of change of f(x) at each point on its graph. In other words, it tells us how steep the graph is at any given point. Since f(x) is increasing, the slope of its graph is positive. And since f'(x) represents the slope of the graph, it follows that f'(x) is positive when f(x) is increasing. In summary, if f(x) is increasing, then f'(x) will be positive.

More Answers: