limit as h approaches 0 of [f(x+h)-f(x)]/h
Formal definition of derivative
The expression [f(x+h) – f(x)]/h represents the slope of the line passing through the points (x, f(x)) and (x+h, f(x+h)). The limit as h approaches 0 represents the instantaneous rate of change of the function f(x) at the point (x, f(x)).
This is known as the derivative of the function f(x) at the point x and is denoted by f'(x). Thus, we can say that
f'(x) = lim h -> 0 [f(x+h) – f(x)]/h
This expression is known as the first principle of differentiation and is a fundamental concept in calculus. The derivative of a function at a point gives us information about the behavior of the function in the vicinity of that point.
In practice, we can use this expression to find the derivative of a function at a given point by evaluating the limit. Alternatively, we can use techniques such as the power rule, product rule or chain rule to find the derivative of more complex functions.
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