How to get the derivative of ( ln &#x2061;<!-- ⁡ --> ( x ) ) <mrow

Lucille Cummings

Lucille Cummings

Answered question

2022-06-21

How to get the derivative of ( ln ( x ) ) sec ( x ) ?
? I know that the derivative of ln ( x ) is 1 x but what happens when you take it to an exponent of sec ( x )?

Answer & Explanation

Colin Moran

Colin Moran

Beginner2022-06-22Added 21 answers

Note that
( ln x ) sec x = ( e ln ln x ) sec x = e ( ln ln x ) ( sec x ) ;
now differentiate it as you would any exponential, not forgetting to use the chain rule.
enrotlavaec

enrotlavaec

Beginner2022-06-23Added 3 answers

Let
y = ln ( x ) sec ( x )
Now sse implicit differentiation to find the derivative
ln y = ln ( ln ( x ) sec ( x ) )
ln y = sec ( x ) ln ( ln ( x ) )
d d x ( ln y ) = d d x ( sec ( x ) ln ( ln ( x ) ) )
1 y d y d x = d d x ( 1 cos ( x ) ln ( ln ( x ) ) )
1 y d y d x = d d x ( 1 cos ( x ) ) ( ln ( ln ( x ) ) ) + d d x ( ( ln ( ln ( x ) ) ) ) 1 cos ( x )
1 y d y d x = sec ( x ) tan ( x ) ( ln ( ln ( x ) ) ) + 1 y 1 ln ( x ) 1 cos ( x )
1 y d y d x = sin ( x ) cos 2 ( x ) ln ( ln ( x ) ) + 1 x ln ( x ) cos ( x )
d y d x = ln ( x ) sec ( x ) ( sin ( x ) ln ( ln ( x ) ) cos 2 ( x ) + 1 x ln ( x ) cos ( x ) )

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