Calculus/Derivatives

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Derivative of a function at a number [edit]

Notation[edit]

We denote the derivative of a function at a number as .

Definition[edit]

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


An analagous equation can be defined by letting . Then , which shows that when approaches , approaches :


Interpretations[edit]

As the slope of a tangent line[edit]

Given a function , the derivative can be understood as the slope of the tangent line to at :

Graph of parabola and tangent line.png
Example[edit]

Find the equation of the tangent line to at .

Solution[edit]

To find the slope of the tangent, we let and use our first definition:


It can be seen that as approaches , we are left with . If we plug in for :


So our preliminary equation for the tangent line is . By plugging in our tangent point to find , we can arrive at our final equation:


So our final equation is .

As a rate of change[edit]

The derivative of a function at a number can be understood as the instantaneous rate of change of when .

The derivative as a function[edit]

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

Notation[edit]

Likewise to the derivative of at , the derivative of the function is denoted .

Definition[edit]

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

Also,

or


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