Differential calculus - derivatives
      Derivatives of basic or elementary functions
         The derivative of the sine function
         The derivative of the cosine function
         The derivative of the exponential function
         The derivative of the logarithmic function
We use the limit definition 
to find the derivative of a function.
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    
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    
Calculus contents C
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