Understanding the Concept of Positive Derivative: Exploring the Relationship between f ‘(x) and the Increasing Function f(x)

When f ‘(x) is positive, f(x) is

When f ‘(x) is positive, it means that the derivative of the function f(x) with respect to x is positive

When f ‘(x) is positive, it means that the derivative of the function f(x) with respect to x is positive. In other words, the function f(x) is increasing in the interval where f ‘(x) is positive.

To understand this concept visually, imagine plotting the graph of the function f(x) on a coordinate plane. When f ‘(x) is positive, it implies that the slope of the tangent line to the graph of f(x) at any given point in the interval is positive. This means that as x increases, the y-values of f(x) also increase.

Let’s consider an example to further illustrate this. Suppose we have a function f(x) = x^2. We can find its derivative f ‘(x) by applying the power rule of differentiation, which states that if f(x) = x^n, where n is a constant, then f ‘(x) = nx^(n-1).

Applying the power rule to f(x) = x^2, we get f ‘(x) = 2x. Now, we can observe that f ‘(x) = 2x is positive for x > 0, which means that f(x) = x^2 is increasing for x > 0. This aligns with our visual understanding as well because when we plot the graph of f(x) = x^2, we see that the curve starts at the origin and moves upward as x increases, indicating an increasing function.

In summary, when f ‘(x) is positive, it indicates that f(x) is increasing in the respective interval.

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