f(x)=\\frac{\\sqrt{2x^{2}}}{\\cos(x)}\nwhich sequence of rules can be used in order to differentiate f\nin…

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

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

Answer

Explanation:

Step1: Analyze the function form

The function ( f(x)=\frac{\sqrt{2x^{2}}}{\cos(x)} ) is a quotient of two functions ( u(x)=\sqrt{2x^{2}}=(2x^{2})^{\frac{1}{2}} ) and ( v(x)=\cos(x) ). So, we first use the quotient rule. The quotient rule states that if ( y = \frac{u}{v} ), then ( y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}} ).

Step2: Differentiate ( u(x)=(2x^{2})^{\frac{1}{2}} )

For ( u(x)=(2x^{2})^{\frac{1}{2}} ), we use the chain rule. Let ( t = 2x^{2} ), then ( u(t)=t^{\frac{1}{2}} ). The chain rule states that if ( y = f(g(x)) ), then ( y^\prime=f^\prime(g(x))\cdot g^\prime(x) ). ( u^\prime(t)=\frac{1}{2}t^{-\frac{1}{2}} ) and ( t^\prime = 4x ), so ( u^\prime(x)=\frac{1}{2}(2x^{2})^{-\frac{1}{2}}\cdot4x). ( v^\prime(x)=-\sin(x) ) (derivative of ( \cos(x) ) is ( -\sin(x) )).

Answer:

D. Quotient rule, then chain rule