Understanding the Derivative | Notation and Calculations

1. f'(x)

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

The notation f'(x) represents the derivative of a function f with respect to the variable x. In other words, it represents the rate at which the function f is changing at a specific point x. The derivative of a function gives us information about its slope or rate of change.

To find the derivative of a function, we can use the rules of differentiation. For example, if f(x) = x^2, we can find f'(x) by using the power rule for derivatives. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Using this rule, we can compute the derivative of f(x) = x^2 as follows:
f'(x) = 2*x^(2-1) = 2*x

Therefore, f'(x) = 2*x for the function f(x) = x^2.

The derivative can also be interpreted geometrically as the slope of the tangent line to the graph of the function at a specific point. It provides valuable information about the behavior of the function, such as the location of extrema (maximum and minimum points) and the concavity of the graph.

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