question_answer

Differentiate log sin x with respect to \[\sqrt{\cos x}.\]

Answer:

Let u = log sin x and \[v=\sqrt{\cos \,x.}\] Then, \[\frac{du}{dx}=\cot \,\,x\] and       \[\frac{dv}{dx}=-\frac{\sin \,x}{2\sqrt{\cos \,x}}\] \[\therefore \]      \[\frac{du}{dv}=\frac{du/dx}{dv/dx}=-\frac{\cot x}{\frac{\sin x}{2\sqrt{\cos \,x}}}\]             \[=-\,2\sqrt{\cos \,x}\cot \,x\,\text{cosec}\,x\]