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telegrafyx

telegrafyx

Answered question

2022-06-15

Evaluate d d x { ( sin x ) cos x + ( cos x ) sin x } with logarithmic differentiation
Spivak asks us to evaluate
d d x { ( sin x ) cos x + ( cos x ) sin x }
by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to be cumbersome), or is there something else I'm missing?

Answer & Explanation

Braedon Rivas

Braedon Rivas

Beginner2022-06-16Added 24 answers

Hint. Take f ( x ) = ( sin x ) cos x . You try to differentiate f ( x ) + f ( π / 2 x ). Use logarithmic differentiation to derivative f ( x )
Jackson Duncan

Jackson Duncan

Beginner2022-06-17Added 10 answers

Let f ( x ) = u + v where u = ( sin x ) cos x and v = ( cos x ) sin x
As we know, If f ( x ) = u + v , f ( x ) = d u d x + d v d x
Consider u = ( sin x ) cos x
ln u = cos x ln ( sin x )
Differentiating, 1 u d u d x = cos x cos x sin x sin x ln ( sin x ) --- (A). by product rule
Now do similarly for d v d x and find d u d x + d v d x = f ( x )
Edit: You will have to multiply (A) by u to get d u d x

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