What is the derivative of cot^2(sin x)?

muldie64bp

muldie64bp

Answered question

2023-03-22

What is the derivative of cot 2 ( sin x ) ?

Answer & Explanation

Makaila Giles

Makaila Giles

Beginner2023-03-23Added 3 answers

Explanation:
Using the product rule, find the derivative.
Chain rule: given a function f ( x ) = g ( h ( x ) ) , d f d x = h ( x ) g ( h ) .
Product Rule: when asked to differentiate a function of the form f ( x ) = g ( x ) h ( x ) , d f d x = g ( x ) h ( x ) + g ( x ) h ( x )
To apply the product rule, we will phrase the problem as follows:
f ( x ) = cot ( sin x ) cot ( sin x )
Thus,
d f d x = ( d d x ( cot ( sin x ) ) ) cot ( sin x ) + cot ( sin x ) ( d d x ( cot ( sin x ) ) ) = 2 cot ( sin x ) d d x cot ( sin x )
Further, we know d d x sin x = cos x . Therefore if our function is cot (sin x), our derivative will take the form cos ( x ) ( - csc 2 ( sin x ) )
Therefore, our overal, derivative is;
d f d x = 2 cot ( sin x ) ( ( cos x ) ( - csc 2 ( sin x ) ) ) = - 2 csc 2 ( sin x ) ( cot ( sin x ) ) ( cos ( x ) )

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