Limit Definition of Derivative
limit (as h approaches 0)= F(x+h)-F(x)/h
The limit definition of derivative is a mathematical definition used to describe the instantaneous change of a function at a specific point. The derivative of a function f(x) at a point x is given by the following formula:
f'(x) = lim h→0 (f(x+h) – f(x))/h
In this formula, h represents the change in x and is called the increment. The limit as h approaches 0 is used to describe the instantaneous change in the function at the point x. The derivative essentially describes the slope of the tangent line at a particular point on the function.
For example, let’s say we have a function f(x) = x². To find the derivative of this function at x = 3, we would plug in the values into the limit definition formula, like this:
f'(3) = lim h→0 (f(3+h) – f(3))/h
f'(3) = lim h→0 ((3+h)² – 3²)/h
f'(3) = lim h→0 (9+6h+h² – 9)/h
f'(3) = lim h→0 (6h+h²)/h
f'(3) = lim h→0 (h(6+h))/h
f'(3) = lim h→0 (6+h)
f'(3) = 6
Therefore, the derivative of f(x) = x² at x = 3 is 6. This tells us that at the point (3, 9) on the function, the instantaneous rate of change (or slope) is equal to 6.
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