h(x)=\\frac{\\cos^{2}(x)}{x}\nwhich sequence of rules can be used in order to differentiate h in its current…

h(x)=\\frac{\\cos^{2}(x)}{x}\nwhich sequence of rules can be used in order to differentiate h in its current form?\nchoose 1 answer:\na product rule, then quotient rule\nb quotient rule, then quotient rule again\nc quotient rule, then chain rule\nd chain rule, then chain rule again

h(x)=\\frac{\\cos^{2}(x)}{x}\nwhich sequence of rules can be used in order to differentiate h in its current form?\nchoose 1 answer:\na product rule, then quotient rule\nb quotient rule, then quotient rule again\nc quotient rule, then chain rule\nd chain rule, then chain rule again

Answer

Explanation:

Step1: Analyze the function form

The function ( h(x)=\frac{\cos^{2}(x)}{x} ) is a quotient of two functions ( f(x)=\cos^{2}(x) ) and ( g(x) = x ). So, the quotient rule (\left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}}) is applicable first. Here ( u=\cos^{2}(x) ) and ( v = x ).

Step2: Differentiate (u=\cos^{2}(x))

To find (u'=\frac{d}{dx}(\cos^{2}(x))), we use the chain rule. Let ( y = u^{2}) with (u=\cos(x)). By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). (\frac{dy}{du} = 2u) and (\frac{du}{dx}=-\sin(x)), so (\frac{d}{dx}(\cos^{2}(x))=2\cos(x)(-\sin(x)))

Answer:

C. Quotient rule, then chain rule