Calculus/Derivatives

< Calculus

Derivative of a function f at a number a

Notation

We denote the derivative of a function f at a number a as f'(a)\,\!.

Definition

The derivative of a function f at a number a a is given by the following limit (if it exists):

f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}


An analagous equation can be defined by letting x=(a+h). Then h=(x-a), which shows that when x approaches a, h approaches 0:

f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}


Interpretations

As the slope of a tangent line

Given a function y=f(x)\,\!, the derivative f '(a)\,\! can be understood as the slope of the tangent line to f(x) at x=a:

Example

Find the equation of the tangent line to y=x^2 at x=1.

Solution

To find the slope of the tangent, we let y=f(x) and use our first definition:

f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}\Rightarrow \lim_{h\rightarrow 0}\frac{{\color{Blue}(a+h)^2-(a)^2}}{h}\Rightarrow \lim_{h\rightarrow 0}\frac{{\color{Blue}a^2+2ah+h^2-a^2}}{h}\Rightarrow \lim_{h\rightarrow 0}\frac{{\color{Blue}h(2a+h)}}{h}\Rightarrow \lim_{h\rightarrow 0}{\color{Blue}(2a+h)}


It can be seen that as h approaches 0, we are left with f'(a)={\color{Blue}2a}\,\!. If we plug in 1 for a:

f'({\color{Red}1})=2({\color{Red}1})\Rightarrow {\color{Red}2}


So our preliminary equation for the tangent line is y={\color{Red}2}x+b. By plugging in our tangent point (1,1) to find b, we can arrive at our final equation:

{\color{Red}1}=2({\color{Red}1})+b\Rightarrow -1=b


So our final equation is y=2x-1\,\!.

As a rate of change

The derivative of a function f(x) at a number a can be understood as the instantaneous rate of change of f(x) when x=a.

The derivative as a function

So far we have only examined the derivative of a function f at a certain number a. If we move from the constant a to the variable x, we can calculate the derivative of the function as a whole, and come up with an equation that represents the derivative of the function f at any arbitrary x value. For clarification, the derivative of f at a is a number, whereas the derivative of f is a function.

Notation

Likewise to the derivative of f at a, the derivative of the function f(x) is denoted f'(x)\,\!.

Definition

The derivative of the function f is defined by the following limit:

f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}

Also,

f'(x)=\lim_{h\rightarrow x}\frac{f(x)-f(h)}{h-x}

or

f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x-h)}{2h}


Go to the School of Mathematics
This article is issued from Wikiversity - version of the Tuesday, April 12, 2011. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.