Basic Derivatives and the Limit Definition
Calculus is said to be the mathematics of derivation, so to understand Calculus, we must first understand derivatives. So, what is a derivative? A derivative is a function that equals the rate of change (slope) at any point of any base function (a function used to find f(x)), even curved functions, in the form of the slope of a Tangent line, which is a line derived from a base function that touches a single point of the base function.
Find the derivative of f(x)=2x.
To find this derivative, we first must understand the most basic derivative rule, which is:
When f(x)=ax, then f ‘(x)=a
(f ‘(x), which is said as f prime of x. It is more formally known as dy/dx. The reason we write f ‘(x) or dy/dx in a derived equation is because it means that the function it describes is derived from a base function)
Using this rule, we can easily form the derivative of f(x)=2x, which must work out to:
This means that the tangent line of this function, no matter what x equals, will always have a slope of 2 at any point on the graph of the base function 2x.
But why does this method work, you might ask? To answer this, we must use the limit definition of derivatives to solve this problem.
The limit definition begins by finding the limit the base function using this form:
f ‘(x)=lim (f(x+h)-f(x))/h or
(h being an hypothetical variable meaning the change in x that is to be removed later as the function is derived)
For our base equation, the limit definition form would look like:
f ‘(x)=lim (2(x+h)-2x)/h
To solve, we must first simplify the limit form:
Limit Def. form:
f ‘(x)= lim (2(x+h)-2x)/h
First, we spread the integer 2 amongst the two inside variables:
= f ‘(x)= lim (2x+2h-2x)/h
Next, we cancel the two 2x’s out, leaving us with:
=f ‘(x)= lim 2h/h
Dividing the two h variables, we can cancel to get:
=f ‘(x)=lim 2
Solving for the limit of this, since the h variable is completely canceled out of the equation, gives us the answer:
=f ‘(x)= 2
Which is the same answer we got using the derivative rule.
Both of these methods can be used to derive simple equations like this, but the derivative rules, of which there are many, are much easier and more useful to use than the limit definition. The limit definition is typically only used when asked for, as it takes a lot longer to do. In future tutorials, we will go over the other derivative rules, and their limit definition counterparts.
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