Understanding the Relationship Between a Decreasing Function and Its Derivative: Explained Mathematically

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

If f(x) is a decreasing function, it means that as x increases, the corresponding values of f(x) are getting smaller

If f(x) is a decreasing function, it means that as x increases, the corresponding values of f(x) are getting smaller. In other words, the function has a negative slope.

To determine the relationship between f(x) and its derivative f'(x), we can recall that the derivative measures the rate of change of a function at any point. For a decreasing function, the rate of change is negative because the function is becoming smaller as x increases.

Therefore, if f(x) is decreasing, it implies that f'(x) is negative. In mathematical terms, we can say that f'(x) < 0 for all x in the domain of f(x). This means that the derivative function f'(x) is always negative, indicating a downward slope for the tangent line at any point on the graph of f(x). It is important to note that this relationship holds true if f(x) is continuously differentiable, meaning that its derivative f'(x) exists for all x-values in its domain.

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