Derivatives of basic or elementary functions
The derivative of the exponential function
Differentiation rules
The derivative of the sum or difference of two functions
The product rule
A constant times a function rule
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:
Functions contents F