
MATHEMATICS  THEORY



The problems involving rate of changes exist in many area of engineering research. Since this type of limit happens so widely, it is given a special name  derivative. 





Derivatives



The notation of derivative is f^{ '}(a). The derivative of a function
f at a given point a is defined as:
(1)
The definition does assume that the limit exists. In order
to extend this definition, let x = a + h, substitute x into f^{ '}(a)
and will get
(2)
The notation of a derivative can be written as: f^{ '}(a),
y^{ '}, df/dx, dy/dx, Df(x), D_{x}f(x). 





Explanation of the Derivative

Tangent 

A good way to understand derivatives is to think about
a tangent line. According to the definition of the tangent line to
a curve y = f(x) at point A(a,
f(a)), the tangent line can be written as:
(3)
Notice that this definition is the same as the definition of derivative
f^{ '}(a).
In other words, the tangent line to y = f(x) at point A(a, f(a)) is
the line that passes through (a, f(a)) and whose slop is equal to the
derivative of f at a.




Rate of Change 

In a previous
section,
the concept of rate
of change was introduced. Is it related to
derivative? The answer is yes.
In a small interval [x_{1}, x_{2}], the changes in x
is
Δx = x_{2}  x_{1}
The
corresponding change in y is
Δy = y_{2}  y_{1}
The instantaneous rate of change is
According to equation (3), r is the derivative of f(x) at x_{1}






Derivative
Formulas



Calculating derivative
according to its definition is tedious. Some rules have been developed
for finding derivatives without having to use the definition directly. 
F(x) 
F^{ '}(x) 
c 
0 
x^{n} 
nx^{n1} 
cf(x) 
cf^{ '}(x) 
f(x) + g(x) 
f^{ '}(x) + g^{ '}(x) 
f(x)  g(x) 
f^{ '}(x)  g^{ '}(x) 
f(x)g(x) 
f^{ '}(x)g(x) + f^{ }(x)g^{ '}(x) 
f(x)/g(x) 
(f^{ '}(x)g(x)  f^{ }(x)g^{ '}(x))/g^{2}(x) 
x^{n} 
nx^{n1} 


 If F is a constant function, F(x) = c,
then F^{ '}(x) = 0.
 If F(x) = x^{n}, where n is a positive integer,
then F^{ '}(x)
= nx^{n1}.
 Assume that c is constant and f^{ '}(x) and g^{ '}(x)
exist.
 If F(x) = cf(x), then F^{ '}(x) = cf^{ '}(x)
 If F(x) = f(x) + g(x), then F^{ '}(x) =
f^{ '}(x) + g^{ '}(x)
 If F(x) = f(x)  g(x), then F^{ '}(x) = f^{ '}(x) 
g^{ '}(x)
 If F(x) = f(x)g(x), then
F^{ '}(x) = f^{ '}(x)g(x)
+ f^{ }(x)g^{ '}(x)
 If F(x) = f(x)/g(x), then
F^{ '}(x) = (f^{ '}(x)g(x)  f^{ }(x)g^{ '}(x))/g^{2}(x)
 If F(x) = x^{n}, where n is a positive integer, then
F^{ '}(x) = nx^{n1}




