g(x)=6x^{3}\\sqrt{x}\\csc(x)\nwhich sequence of rules can be used in order t its current form?\nchoose 1…

Explanation:

Step1: Recall the product rule

The product rule states that if (y = u\cdot v\cdot w), then (y^\prime=u^\prime vw + uv^\prime w+uvw^\prime). Here (u = 6x^{3}\sqrt{x}=6x^{3 + \frac{1}{2}}=6x^{\frac{7}{2}}), (v = 1) (temporarily considering two - part product first conceptually), (w=\csc(x)). First, we can think of (g(x)) as a product of two functions (y_1=6x^{\frac{7}{2}}) and (y_2 = \csc(x)). But more accurately, if we consider the general form of the product rule for three functions (y=f(x)\cdot g(x)\cdot h(x)), we can apply the product rule multiple times. If we first consider (u = 6x^{\frac{7}{2}}) and (v=\csc(x)), and then if we had another decomposition (even though (6x^{\frac{7}{2}}) is a single - term power function, the product rule for differentiation of (y = a(x)\cdot b(x)\cdot c(x)) is (y^\prime=a^\prime(x)b(x)c(x)+a(x)b^\prime(x)c(x)+a(x)b(x)c^\prime(x)), which is equivalent to applying the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u=a(x)b(x)) and (v = c(x)) (first apply product rule to (u=a(x)b(x)) as (u^\prime=a^\prime(x)b(x)+a(x)b^\prime(x)), then ((uv)^\prime=(a^\prime(x)b(x)+a(x)b^\prime(x))c(x)+a(x)b(x)c^\prime(x)))

Step2: Analyze the chain - rule

The chain rule is (y = f(g(x))), (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). The function (g(x)=6x^{3}\sqrt{x}\csc(x)) is a product of functions (y_1 = 6x^{\frac{7}{2}}) (a power function, derivative using power rule (y^\prime=ax^{n}), (y^\prime=anx^{n - 1})) and (y_2=\csc(x)) (where (\csc(x)=\frac{1}{\sin(x)}), and if we consider (y = \csc(x)) as (y=( \sin(x))^{-1}), using the chain rule (y^\prime=-1\times(\sin(x))^{-2}\cdot\cos(x)=-\cot(x)\csc(x))). But the main operation for differentiation of (g(x)) is the product rule (since (g(x)) is a product of functions of (x), not a composition in the sense required for the chain - rule as the first step). The chain rule is used for composite functions (y = f(g(x))), but here (g(x)) is a product (u(x)\cdot v(x)) where (u(x)=6x^{\frac{7}{2}}) and (v(x)=\csc(x)). To differentiate (g(x)), we first use the product rule ((uv)^\prime=u^\prime v + uv^\prime). There is no need for the chain rule as the first - step operation. If we assume (g(x)) is a product of three functions (a(x)=6x^{3}), (b(x)=\sqrt{x}), (c(x)=\csc(x)), ((a(x)b(x)c(x))^\prime=(a(x)b(x))^\prime c(x)+a(x)b(x)c^\prime(x)=(a^\prime(x)b(x)+a(x)b^\prime(x))c(x)+a(x)b(x)c^\prime(x)), which is two applications of the product rule

Answer:

A. Product rule, then product rule again