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

write the following function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x.\n$y = (\frac{x^{2}}{6}+3x - \frac{4}{x})^{6}$\nwrite the given function in the form y = f(u) and u = g(x). choose the correct answer below.\na. y = f(u)=u^{6} and u = g(x)=(\frac{x^{2}}{6}+3x - \frac{4}{x})^{6}\nb. y = f(u)=u^{6} and u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x}\nc. y = f(u)=u and u = g(x)=\frac{x^{2}}{6}+3x\nd. y = f(u)=u and u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x}$

write the following function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x.\n$y = (\frac{x^{2}}{6}+3x - \frac{4}{x})^{6}$\nwrite the given function in the form y = f(u) and u = g(x). choose the correct answer below.\na. y = f(u)=u^{6} and u = g(x)=(\frac{x^{2}}{6}+3x - \frac{4}{x})^{6}\nb. y = f(u)=u^{6} and u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x}\nc. y = f(u)=u and u = g(x)=\frac{x^{2}}{6}+3x\nd. y = f(u)=u and u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x}$

Answer

Explanation:

Step1: Identify f(u) and g(x)

We have a composite - function. Let (u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x}) and (y = f(u)=u^{6}). This is because the original function (y=\left(\frac{x^{2}}{6}+3x - \frac{4}{x}\right)^{6}) can be thought of as first evaluating the expression (\frac{x^{2}}{6}+3x - \frac{4}{x}) (which is (u)) and then raising it to the 6 - th power (which is (y = f(u))).

Step2: Find (\frac{dy}{du}) and (\frac{du}{dx})

Differentiate (y = f(u)=u^{6}) with respect to (u). Using the power - rule (\frac{d}{du}(u^{n})=nu^{n - 1}), we get (\frac{dy}{du}=6u^{5}). Differentiate (u = g(x)=\frac{x^{2}}{6}+3x - 4x^{-1}) with respect to (x). Using the power - rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), we have (\frac{du}{dx}=\frac{2x}{6}+3+4x^{-2}=\frac{x}{3}+3+\frac{4}{x^{2}}).

Step3: Use the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})

Substitute (u = \frac{x^{2}}{6}+3x - \frac{4}{x}) into (\frac{dy}{du}) and multiply by (\frac{du}{dx}). (\frac{dy}{dx}=6\left(\frac{x^{2}}{6}+3x - \frac{4}{x}\right)^{5}\left(\frac{x}{3}+3+\frac{4}{x^{2}}\right))

Answer:

B. (y = f(u)=u^{6}) and (u = g(x)=\frac{x^{2}}{6}+3x - \frac{4}{x})