Understanding the Alternate Definition of the Derivative | A Powerful Tool for Calculating Instantaneous Rate of Change

Alternate Definition of Derivative

The alternate definition of the derivative is given by the concept of a difference quotient

The alternate definition of the derivative is given by the concept of a difference quotient. Let’s define a function f(x) that is differentiable at a point x = a. The derivative of f(x) at x = a, denoted as f'(a) or dy/dx|a, can alternatively be defined as follows:

f'(a) = lim┬(h→0)⁡(f(a+h) – f(a))/h

Here, h is a small value that approaches zero. By taking the limit as h approaches zero, we can determine the instantaneous rate of change of the function at x = a. The numerator (f(a+h) – f(a)) represents the change in the value of the function, and dividing it by h gives us the average rate of change over a small interval. As h approaches zero, the average rate of change becomes the exact rate of change at x = a.

This alternate definition of the derivative allows us to compute the derivative of a function by evaluating the limit of a difference quotient. It provides a numerical approximation of the slope of the tangent line to the graph of the function at x = a.

Using this definition, we can find the derivative of various functions, including polynomial functions, trigonometric functions, exponential functions, etc., by substituting the function into the difference quotient and evaluating the limit.

It’s important to note that the alternate definition of the derivative is equivalent to the standard definition in terms of limits, which involves using the power rule, product rule, chain rule, etc., to differentiate specific types of functions. Both definitions provide the same result and are used interchangeably depending on the situation.

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