How do you find the derivative of y=ln|sec x+tan x|?

Evelyn Buchanan

Evelyn Buchanan

Answered question

2023-03-11

How to find the derivative of y = ln | sec x + tan x | ?

Answer & Explanation

Kayden Yates

Kayden Yates

Beginner2023-03-12Added 3 answers

                              y   =   sec x .
Solution:
We'll use the Chain Rule to accomplish this.
Recall:                y   =   ln | x |           y   =   1 x .
Therefore, by the Chain Rule:
        y   =   ln | f ( x ) |           y   =   [ 1 f ( x ) ] f ( x )   =   f ( x ) f ( x ) .
               y   =   ln | f ( x ) |           y   =   f ( x ) f ( x ) .
Thus, in our example:
                         y   =   ln | sec x + tan x | ;                f ( x )   =   sec x + tan x
                       y   =   ( sec x + tan x ) sec x + tan x ;              =   f ( x ) f ( x )
                              =   ( sec x ) + ( tan x ) sec x + tan x ;
                              =   sec x tan x + sec 2 x sec x + tan x ;
                              =   sec x ( tan x + sec x ) sec x + tan x ;
                              =   sec x ( sec x + tan x ) sec x + tan x ;
                              =   sec x ( sec x + tan x ) ( sec x + tan x ) ; #
                              =   sec x .
Thus, we have now:
                                                     y   =   sec x .
Summarizing:
                    y   =   ln | sec x + tan x |           y   =   sec x .

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