Chapter 9: Derivative of Trig Functions

To find the slope of the tangent line to the curve y = , we must go back to the definition of the derivative as the limit of the difference quotient:

= .

Recall the procedure used to find the derivative of the much simpler function : expand , simplify algebraically, and then take the limit as gets closer and closer to 0. We do essentially the same thing for , but we need the following three basic facts in order to do this:

Fact 1. ;

Fact 2. ;

Fact 3. .

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You know the exact value . Find using the tangent line approximation , and compare with the `exact' value given by calculator.


In the tangent line approximation we choose , = and = . Here it was important that we changed from degrees to radians. By the tangent line approximation,

= (approx) 0.5+ = 0.5+ = 0.5+ ;

the exact value is 0.515038 ....

When we have a derivative formula for a new function, we can combine it with other rules to find the derivative of a more complicated function that is built up from the basic function.

Find the derivative of ( ).

(Note: (something) means ( ).

To evaluate this derivative, we have to use the chain rule twice:

= ( ) =

cos = ( )
= cos ( ) .

Notation: If f(x) is a function of x then we will also denote by or

All of the other trig functions can be expressed in terms of the sine, and so their derivatives can easily be calculated. The most important ones are:

= = ;

= = =

= = = .