Understanding the concept of a negative derivative | Decreasing Functions and Negative Slope

When f ‘(x) is negative, f(x) is

When the derivative of a function, f ‘(x), is negative, it means that the function is decreasing at that particular point, or in other words, the output values of the function, f(x), are decreasing as the input values, x, increase

When the derivative of a function, f ‘(x), is negative, it means that the function is decreasing at that particular point, or in other words, the output values of the function, f(x), are decreasing as the input values, x, increase.

To understand this concept better, let’s consider an example. Suppose we have a function f(x) = x^2. To find the derivative, f ‘(x), we can apply the power rule of differentiation, which states that the derivative of x^n, where n is a constant, is nx^(n-1).

Therefore, taking the derivative of f(x) = x^2, we have:
f ‘(x) = 2x.

Now, let’s analyze the sign of f ‘(x) in different scenarios:

1. If x > 0: When x is positive, f ‘(x) = 2x is also positive. This means that the function is increasing as x increases. In other words, for every positive value of x, the corresponding output values, f(x), will increase.

2. If x < 0: When x is negative, f '(x) = 2x is negative. This indicates that the function is decreasing as x decreases. So, for every negative value of x, the corresponding output values, f(x), will decrease. Hence, when f '(x) is negative, it means that the function, f(x), is decreasing at that particular point, indicating a negative slope.

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