Understanding Derivatives: Definition and Computation Methods in Mathematics

Alternate definition of derivative

In mathematics, the derivative is a fundamental concept that measures the rate at which a function changes as its input varies

In mathematics, the derivative is a fundamental concept that measures the rate at which a function changes as its input varies. The derivative of a function at a specific point can be thought of as the slope of the tangent line to the graph of the function at that point.

The alternate definition of the derivative is based on the limit of the difference quotient. Given a function f(x), the derivative of f(x) at a point x = a can be defined using the following limit:

f'(a) = lim(h->0) [f(a+h) – f(a)] / h

In this alternate definition, h represents a small change in the input variable x. By taking the limit as h approaches zero, we can determine the instantaneous rate of change of the function at the specific point a.

To compute the derivative using this definition, we substitute the function values f(a+h) and f(a) into the expression [f(a+h) – f(a)] / h and evaluate the limit as h approaches zero. This process involves simplifying the expression and cancelling terms to simplify further until the limit can be evaluated. The resulting value gives the derivative of the function at the point a.

It is worth mentioning that this alternate definition is entirely equivalent to the more commonly used definition of the derivative as the limit of the difference quotient as h approaches zero. Both definitions yield the same derivative value, and they are interchangeable depending on the context and convenience.

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