Question

write the following function in the form ( y = f(u) ) and ( u = g(x) ). then find ( \frac{dy}{dx} ) as a function of ( x ).

( y=left(\frac{x^{2}}{4}+3 x-\frac{2}{x}
ight)^{5} )

write the given function in the form ( y = f(u) ) and ( u = g(x) ). choose the correct answer below.

a. ( y = f(u)=u^{5} ) and ( u = g(x)=\frac{x^{2}}{4}+3 x-\frac{2}{x} )
b. ( y = f(u)=u ) and ( u = g(x)=\frac{x^{2}}{4}+3 x )
c. ( y = f(u)=u^{5} ) and ( u = g(x)=left(\frac{x^{2}}{4}+3 x-\frac{2}{x}
ight)^{5} )

Explanation:

Step1: Analyze the function structure

The given function \(y = (\frac{x^{2}}{4}+3x - \frac{2}{x})^{5}\) is a composite function. We can let \(u=\frac{x^{2}}{4}+3x-\frac{2}{x}\), and then \(y = u^{5}\).

Step2: Check each option

• Option A: If \(y = f(u)=u^{5}\) and \(u = g(x)=\frac{x^{2}}{4}+3x-\frac{2}{x}\), this matches the composite - function structure \(y=(g(x))^{5}\).
• Option B: If \(y = f(u)=u\) and \(u = g(x)=\frac{x^{2}}{4}+3x\), then \(y=\frac{x^{2}}{4}+3x\), which is not the given function \(y = (\frac{x^{2}}{4}+3x - \frac{2}{x})^{5}\).
• Option C: If \(y = f(u)=u^{5}\) and \(u = g(x)=(\frac{x^{2}}{4}+3x - \frac{2}{x})^{5}\), then \(y = u^{5}=((\frac{x^{2}}{4}+3x - \frac{2}{x})^{5})^{5}=(\frac{x^{2}}{4}+3x - \frac{2}{x})^{25}\), which is not the given function.

Answer:

A. \(y = f(u)=u^{5}\) and \(u = g(x)=\frac{x^{2}}{4}+3x-\frac{2}{x}\)