Understanding Increasing Functions and Positive Derivatives in Mathematics

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, the function is getting “steeper” or “higher” as x increases.

The derivative of a function, denoted as f'(x) or dy/dx, represents the rate of change of the function with respect to x. It tells us how fast the function is changing at a particular point.

Therefore, if f(x) is increasing, we can conclude that f'(x) must be positive. This is because a positive derivative value indicates that the function is increasing at that point.

To further clarify this concept, let’s consider an example:

Let’s say we have a function f(x) = 2x^2. We can find the derivative of this function using the power rule of differentiation:

f'(x) = 2 * 2x^(2-1)
= 4x

Now, let’s analyze the increasing nature of the function and its derivative. If we plot the graph of f(x) = 2x^2, we will observe a parabola that opens upward.

As x increases, the values of f(x) also increase. This confirms that the function f(x) = 2x^2 is indeed increasing.

If we analyze the derivative f'(x) = 4x, we can see that it is always positive. Hence, the derivative of the increasing function f(x) = 2x^2 is positive, indicating the increasing nature of the function.

In summary, if a function f(x) is increasing, its derivative f'(x) will be positive.

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