Differential calculus
      Differential Calculus - Derivatives and differentials
      The derivative of a function
         Definition of the derivative of a function
      Tangent to a curve
      The equation of the line tangent to the given curve at the given point
         Determining the derivative of a function as the limit of the difference quotient
         The equation of the line tangent to the given curve at the given point example
The derivative of a function
The slope or the gradient of the secant line joining points (x, f (x)) and (x + Dx, f (x + Dx)) given by 
is called the difference quotient, and represents the average rate of change of the function y = f (x).
Note that in the above equation x is the given fixed point and the only variable quantity is the distance increment, Dx.
Definition of the derivative of a function
As the distance Dx gets smaller, the average rate of change becomes more accurate, that is the secant line becomes more and more like the tangent line of the curve at the point (x, f (x)), as shows the figure above.
Therefore, the limit of the difference quotient as  Dx 0 that equals the slope of the line tangent to the curve at the point (x, f (x)),
represents the instantaneous rate of change of the function with respect to x, or when written as
   
where Dx is substituted with h, is called the derivative of a function f at a given point x.
Geometrically, the derivative is the slope m (gradient) of the tangent at the point (x, y) to the curve y = f (x).
There are other common notations for the derivative like,  
The equation of the line tangent to the given curve at the given point
The equation of the tangent line at the given point (x0, y0) of a function y = f (x) we write using the point slope equation of a line,
y - y0 = f ' (x0) (x - x0)
where  m = f ' (x0) is the slope of the tangent line at the point (x0, y0).
Determining the derivative of a function as the limit of the difference quotient
Example:  Find the equation of the tangent line at the point (2, 4) of the function  f (x) = x2.
Solution:  The slope of the tangent line at the given point we calculate using the formula
then,  
m = f ' (x0) = 2x0,   m = f ' (2) = 2 2 = 4.
By substituting  m = 4 and the point (2, 4) into
y - y0 = f ' (x0) (x - x0)
we get
 y -  4 = 4 (x - 2)   or   y = 4x -
the equation of the tangent line at the given point of the function  f (x) = x2, as shows the right figure.
 
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