differentiate the following function\n$y = \\frac{x^{6}\\ln x}{6}$\nlet $f(x)=\\frac{x^{6}}{6}$ and…

differentiate the following function\n$y = \\frac{x^{6}\\ln x}{6}$\nlet $f(x)=\\frac{x^{6}}{6}$ and $g(x)=\\ln x$. choose the correct setup below to start differentiating the function.\n$\\bigcirc$ a. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x+\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x$\n$\\bigcirc$ b. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x+\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x$\n$\\bigcirc$ c. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x+\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x$\n$\\bigcirc$ d. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\frac{d}{dx}\\ln x$\n$\\frac{d}{dx}\\left(\\frac{x^{6}\\ln x}{6}\\right)=\\square$\n

differentiate the following function\n$y = \\frac{x^{6}\\ln x}{6}$\nlet $f(x)=\\frac{x^{6}}{6}$ and $g(x)=\\ln x$. choose the correct setup below to start differentiating the function.\n$\\bigcirc$ a. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x+\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x$\n$\\bigcirc$ b. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x+\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x$\n$\\bigcirc$ c. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\ln x+\\frac{x^{6}}{6}\\cdot\\frac{d}{dx}\\ln x$\n$\\bigcirc$ d. $\\frac{d}{dx}\\left\\frac{x^{6}\\ln x}{6}\\right=\\frac{d}{dx}\\left\\frac{x^{6}}{6}\\right\\cdot\\frac{d}{dx}\\ln x$\n$\\frac{d}{dx}\\left(\\frac{x^{6}\\ln x}{6}\\right)=\\square$\n

Answer

Explanation:

Step1: Apply the product rule

The product rule states that if (y = f(x)g(x)), then (y^\prime=f^\prime(x)g(x)+f(x)g^\prime(x)). Here (f(x)=\frac{x^{6}}{6}) and (g(x)=\ln x). So (\frac{d}{dx}\left(\frac{x^{6}\ln x}{6}\right)=\frac{d}{dx}\left(\frac{x^{6}}{6}\right)\cdot\ln x+\frac{x^{6}}{6}\cdot\frac{d}{dx}[\ln x])

Step2: Differentiate (\frac{x^{6}}{6}) and (\ln x)

Using the power rule (\frac{d}{dx}(x^{n}) = nx^{n - 1}), (\frac{d}{dx}\left(\frac{x^{6}}{6}\right)=\frac{6x^{5}}{6}=x^{5}). Using the formula (\frac{d}{dx}(\ln x)=\frac{1}{x})

Step3: Simplify the expression

Substitute the derivatives into the product - rule formula: (\frac{d}{dx}\left(\frac{x^{6}\ln x}{6}\right)=x^{5}\ln x+\frac{x^{6}}{6}\cdot\frac{1}{x}=x^{5}\ln x+\frac{x^{5}}{6}=x^{5}\left(\ln x+\frac{1}{6}\right))

Answer:

(x^{5}\left(\ln x+\frac{1}{6}\right))