The Relationship Between Increasing Functions and Positive Derivatives | A Comprehensive Explanation

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

If a function f(x) is increasing, it means that as x increases, the corresponding values of f(x) also increase

If a function f(x) is increasing, it means that as x increases, the corresponding values of f(x) also increase. In other words, as you move from left to right along the x-axis, the function goes up.

Now, let’s consider the derivative f'(x) of the function f(x). The derivative f'(x) represents the rate at which the function is changing at a particular point x. If f(x) is increasing, it implies that the derivative f'(x) is positive.

To understand why this is the case, think about the relationship between the slope of a function and its increasing or decreasing behavior. When the function is increasing, the slope (or derivative) is positive, indicating a positive rate of change. On the other hand, when the function is decreasing, the slope (or derivative) is negative, indicating a negative rate of change.

In summary, if a function f(x) is increasing, then the derivative f'(x) is positive.

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