Mastering Second Derivatives: Understanding f”(x) and How to Calculate It

f”(x)

The notation f”(x) represents the second derivative of a function f with respect to the variable x.

The notation f”(x) represents the second derivative of a function f with respect to the variable x. It means taking the derivative of the derivative of the function f(x) with respect to x.

To calculate f”(x), we need to follow these steps:

Step 1: Find the first derivative, f'(x)
Take the derivative of the function f(x) with respect to x. This involves applying the derivative rules, such as the power rule, product rule, and chain rule, if necessary.

Step 2: Find the second derivative, f”(x)
Take the derivative of the first derivative, f'(x), obtained in Step 1, with respect to x. Again, this involves applying the derivative rules as needed.

Overall, finding f”(x) involves differentiating the function twice.

Let’s work through an example:

Example:
Consider the function f(x) = 2x^3 – 5x^2 + 3x – 1. Find f”(x).

Step 1: Find the first derivative, f'(x)
To find the first derivative, we differentiate each term with respect to x using the power rule:

f'(x) = 6x^2 – 10x + 3.

Step 2: Find the second derivative, f”(x)
Now, we differentiate the first derivative, f'(x), with respect to x:

f”(x) = d/dx (6x^2 – 10x + 3)
= 12x – 10.

Therefore, the second derivative of f(x) is f”(x) = 12x – 10.

Note: The second derivative represents the rate of change of the slope of the original function, f(x), with respect to x.

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