Limit Definition of Derivative
The limit definition of the derivative is a fundamental concept in calculus that defines how to calculate the rate of change of a function at a specific point
The limit definition of the derivative is a fundamental concept in calculus that defines how to calculate the rate of change of a function at a specific point. Mathematically, the limit definition of the derivative is given by:
f'(x) = lim(h -> 0) [f(x + h) – f(x)] / h
In this formula, f'(x) represents the derivative of the function f(x) at the point x. The limit as h approaches 0 means that we are considering smaller and smaller values of h to calculate the instantaneous rate of change.
To use the limit definition of the derivative, start by selecting a point on the function, let’s say x=a. Then, compute the limit of the difference quotient as h approaches 0. The result of this limit will give you the rate of change (slope) of the function at that specific point.
Let’s work through an example:
Suppose we have the function f(x) = 3x^2 – 2x + 1 and we want to find the derivative at x=a.
Using the limit definition, we have:
f'(a) = lim(h -> 0) [f(a + h) – f(a)] / h
Plugging in f(x) = 3x^2 – 2x + 1 and simplifying, we get:
f'(a) = lim(h -> 0) [3(a + h)^2 – 2(a + h) + 1 – (3a^2 – 2a + 1)] / h
Expanding and simplifying further, we have:
f'(a) = lim(h -> 0) [3a^2 + 6ah + 3h^2 – 2a – 2h + 1 – 3a^2 + 2a – 1] / h
Canceling out terms and rearranging, we get:
f'(a) = lim(h -> 0) [6ah + 3h^2 – 2h] / h
Factoring out h and canceling, we get:
f'(a) = lim(h -> 0) 6a + 3h – 2
Taking the limit as h approaches 0, we find:
f'(a) = 6a – 2
So, the derivative of f(x) = 3x^2 – 2x + 1 at any point x=a is given by f'(a) = 6a – 2.
This is the general process for finding the derivative of a function using the limit definition. Remember, the limit definition becomes more tedious for more complicated functions, and that’s where other derivative rules and techniques come in handy.
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