Understanding the Second Derivative | Calculating and Applying the Rate of Change of a Function

1. f”(x)

In mathematics, f”(x) refers to the second derivative of a function f with respect to the variable x

In mathematics, f”(x) refers to the second derivative of a function f with respect to the variable x. The second derivative measures the rate of change of the slope of a given function or, in other words, it represents how the rate of change of the function itself changes.

To calculate the second derivative, you need to differentiate the function twice. Let’s say you have a function f(x), and you want to find f”(x). First, differentiate f(x) with respect to x to obtain f'(x), which represents the first derivative. Then, differentiate f'(x) with respect to x again to get f”(x), the second derivative.

For example, consider the function f(x) = 3x^2 + 4x + 2. To find f”(x), we follow these steps:
1. Find the first derivative, f'(x), by applying the power rule of differentiation:
f'(x) = d/dx(3x^2 + 4x + 2) = 6x + 4

2. Find the second derivative, f”(x), by differentiating f'(x) using the power rule:
f”(x) = d/dx(6x + 4) = 6

So, f”(x) = 6 is the second derivative of the given function f(x) = 3x^2 + 4x + 2. This means that the rate of change of the slope of f(x) is a constant value of 6.

The second derivative is useful in analyzing the behavior and properties of functions, such as identifying critical points, finding concavity and inflection points, and determining whether a function has a minimum or maximum point.

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