Question

g(x)=6x^{3}\sqrt{x}\csc(x)
which sequence of rules can be used in order t its current form?
choose 1 answer:
a product rule, then product rule again
b chain rule, then product rule
c product rule, then chain rule
d chain rule, then chain rule again

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