Differentiate the function x sin x with respect to x .

Given x sin x Let y = x sin x d y d x = 1 2 x sin x × d d x ( x sin x ) = 1 2 x sin x × x d sin x d x + sin x d x d x = 1 2 x sin x × ( x cos x + sin x × 1 ) Therefore, d y d x = ( sin x + x cos x ) 2 x sin x

Differentiate the function x sin x with respect to x .

Given x sin x Let y = x sin x d y d x = 1 2 x sin x × d d x ( x sin x ) = 1 2 x sin x × x d sin x d x + sin x d x d x = 1 2 x sin x × ( x cos x + sin x × 1 ) Therefore, d y d x = ( sin x + x cos x ) 2 x sin x

Differentiate cot x with respect to x .

Given cot x Let y = cot x d y d x = 1 2 cot x × d d x cot x = 1 2 cot x × - cosec 2 x × d d x x Therefore, d y d x = - cosec 2 x 4 x cot ( x )

Differentiate cos sin a x + b with respect to x .

d d x cos sin a x + b = - sin sin a x + b × d d x sin a x + b = - sin sin a x + b × cos a x + b × d d x a x + b = - sin sin a x + b × cos a x + b × 1 2 a x + b × d d x a x + b Therefore, d d x cos sin a x + b = - a 2 a x + b cos a x + b · sin sin a x + b