Calculus/Derivatives

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[edit] Derivative of a function f at a number a

[edit] Notation

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

[edit] 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}


[edit] Interpretations

[edit] 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:

Derivative as tangent.jpg
[edit] Example

Find the equation of the tangent line to y = x2 at x = 1.

[edit] 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\,\!.

[edit] 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.

[edit] 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.

[edit] Notation

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

[edit] 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}


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