Defining the Derivative: Learn It 2

The Derivative of a Function at a Point

The type of limit we compute to find the slope of the tangent line to a function at a point has many applications across various disciplines. These include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give it a special name: the derivative. The process of finding a derivative is called differentiation.

derivative

Let [latex]f(x)[/latex] be a function defined in an open interval containing [latex]a[/latex]. The derivative of the function [latex]f(x)[/latex] at [latex]a[/latex], denoted by [latex]f^{\prime}(a)[/latex], is defined by

[latex]f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}[/latex]

provided this limit exists.

 

Alternatively, we may also define the derivative of [latex]f(x)[/latex] at [latex]a[/latex] as

[latex]f^{\prime}(a)=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}[/latex]

provided this limit exists.

For [latex]f(x)=x^2[/latex], estimate [latex]f^{\prime}(3)[/latex] using the first definition of the derivative.

For [latex]f(x)=3x^2-4x+1[/latex], find [latex]f^{\prime}(2)[/latex] by using the second definition.