The derivative as a function
Derivatives of basic or elementary functions
The derivative of the power function
The derivative of the cosine function
The derivative as a function
The derivative of a function f (x) is a new function that contains the value of the derivative of all points on the original function.
That is, the derivative function gives the slope of the line tangent to f (x) at every point x.
The slope of the tangent line provides information about how the graph of the function is changing.
Thus, if        f ' (x) < 0 then  f (x)   is decreasing
and if        f ' (x) > 0 then  f (x)   is increasing,
as is shown in the figure below.
The process of determining or finding the derivative is called differentiation.
In order a function to be differentiable, it must be continuous, and both one-sided limits (the left-handed and right-handed limits) must be equal at the given point.
For example, the function f (x) = | x | is not differentiable at x = 0, although it is continuous there.
Derivatives of basic or elementary functions
Determining the derivative of a function as the limit of the difference quotient
We use the limit definition
to find the derivative of a function.
The derivative of the power function
Given is the power function  f (x) = xn  where n is a a positive integer.
We use the binomial theorem to evaluate  f ( x + h),
So that,   f (x + h) -  f (x) equals,
Then the difference quotient
Since every term except the first is factor of h then, the limit of the difference quotient as h tends to zero is
 Therefore, if then
The derivative of the linear function
Thus, for n = 1, that is for the linear function  f (x) = x, the difference  f (x + h) -  f (x) = x + h - x = h so that the difference quotient equals 1.
 Therefore, if f(x) = x then
The derivative of a constant
The function  f (x) = c, where c is a fixed constant, is graphically represented by a horizontal line so that at any given point (x, f (x)) the slope of the line tangent to the graph of  f  is 0. Therefore,
f(x) = c  and   f(x + h) = c  so that   f(x + h) - f(x) = c - c = 0.
Thus,
 that is, if f(x) = c then
The derivative of the sine function
We use the limit of the difference quotient to find the derivative of the function  f (x) = sin x.
Let rewrite the difference quotient applying the sum to product formula,
Since, the derivative is the limit of the difference quotient as h tends to zero then,
 Therefore, if f(x) = sin x then
The derivative of the cosine function
We use the limit of the difference quotient to find the derivative of the function  f (x) = cos x.
Let rewrite the difference quotient applying the sum to product formula,
Since the derivative is the limit of the difference quotient as h tends to zero then,
 Therefore, if f(x) = cos x then
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