TRIGONOMETRIC FUNCTIONS If we are given
we may state that, from the general formula,
Since
and
Then by substituting equations (5.4) and (5.5) into
equation (5.3),
Now we are interested in finding the derivative
of the function
sin u, so we apply the chain rule
From the chain rule and equation (5.6), we find
In other words, to find the derivative of the sine of a
function, we use the cosine of the function times the derivative of the
function.
By a similar process we find the derivative of the cosine
function to be
_{
}
The derivatives of the other trigonometric functions may
be found by expressing them in terms of the sine and cosine. That is,
and by substituting sin u for u, cos u for v, and du for dx in the
expression of the quotient theorem
we have
Taking
and
and substituting into equation (5.7), we find that
Now using the chain rule and equation (5.8), we find
By stating the other trigonometric functions in terms of
the sine and cosine and using similar processes, we may find the following
derivatives:
EXAMPLE: Find the derivative of the function
SOLUTION.
EXAMPLE. Find the derivative of the function
SOLUTION: Use the power theorem to find
Then find
and
Combining all of these, we find that
PRACTICE PROBLEMS:
Find the derivative of the following:
^{}
ANSWERS:
