Differential calculus - derivatives
Derivatives of basic or elementary functions
The derivative of the cosine function
The derivative of the exponential 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