Derivatives of basic or elementary functions
         Determining the derivative of a function as the limit of the difference quotient
         The derivative of the exponential function
         The derivative of the logarithmic function
      Differentiation rules
         The derivative of the sum or difference of two functions
         The product rule
         A constant times a function rule
         Derivatives of functions, examples
The derivative of the exponential function
Let use the limit of the difference quotient to find the derivative of the function  f (x) = ax.
Since the difference quotient is
then, the derivative as the limit of the difference quotient as h tends to zero
That is, by plugging  t = ah - 1, then  t 0 as  h 0, and
  Therefore, if f(x) = ax  then    
or when the base a is substituted by the natural base e obtained is the exponential function  ex, thus 
   if f(x) = ex  then    
The derivative of the logarithmic function
Let's use the limit of the difference quotient to find the derivative of the function  f (x) = loga x.
The difference quotient applied to the given function
As the derivative is the limit of the difference quotient as h tends to zero, then
Then, applying the base change identity and substituting  a = e
  if    f(x) = loga x then    
  if    f(x) = ln x then    
Differentiation rules
The derivative of the sum or difference of two functions
The derivative of the sum or difference of two differentiable functions equals the sum or difference of their derivatives, written
   
The product rule
The derivative of the product of two differentiable functions is equal to, the first function times the derivative of the second plus the second function times the derivative of the first,
   
A constant times a function rule
The derivative of a constant times a function is equal the constant times the derivative of the function, where the constant c can be any real number or expression that does not contain the variable,
   
Derivatives of functions examples
Example:   Find the derivative of the linear function  f(x) = m x + c.
Solution:    
Example: Find the derivative of the function 
Solution:    
Example: Find the derivative of the function 
Solution:    
Example: Find the derivative of the function 
Solution:    
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