Understanding the Derivative of a Function: Exploring f'(c) and Its Significance

f'(c)

To find the derivative of a function, denoted as f'(x), you need to differentiate the function with respect to the variable x

To find the derivative of a function, denoted as f'(x), you need to differentiate the function with respect to the variable x. However, in the expression f'(c), where c is a constant, the derivative is with respect to x, not c.

If f(x) is a function and c is a constant, then f'(c) can be understood as the value of the derivative of f(x) at the point x = c. In other words, it represents the instantaneous rate of change of the function at that specific point.

To find f'(c), you need to differentiate the function f(x) first, and then evaluate the result at x = c. Let’s denote the derivative of f(x) as f'(x), and the derivative evaluated at c as f'(c).

Example:
Let’s say we have the function f(x) = x^2 + 3x – 2.

Step 1:
Differentiate the function f(x) to find the derivative, f'(x).
Applying the power rule for differentiation, we get:
f'(x) = 2x + 3

Step 2:
Evaluate f'(x) at x = c.
If c is a constant, then f'(c) can be found by substituting c into the derivative expression we found in Step 1.
Therefore, f'(c) = 2c + 3.

Note: The value of f'(c) can be negative, positive, or zero, depending on the specific value of c and the nature of the function f(x).

So, in summary, f'(c) represents the derivative of the function f(x) evaluated at x = c, and it gives the instantaneous rate of change of the function at that point.

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