Introduction to Multivariable Functions: Background You’ll Need 3

  • Find the derivative of a function at a given point

The Derivative of a Function at a Point

The limit we use to find the slope of a tangent line appears so frequently in calculus that it has 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.