compute the derivative. use logarithmic differentiation where appropriate. \n\\frac{d}{dx}\\left(1+\\frac{6}{…

compute the derivative. use logarithmic differentiation where appropriate. \n\\frac{d}{dx}\\left(1+\\frac{6}{x}\\right)^{x}\n\\frac{d}{dx}\\left(1+\\frac{6}{x}\\right)^{x}=\\square\n(use parentheses to clearly denote the argument of each function.)

compute the derivative. use logarithmic differentiation where appropriate. \n\\frac{d}{dx}\\left(1+\\frac{6}{x}\\right)^{x}\n\\frac{d}{dx}\\left(1+\\frac{6}{x}\\right)^{x}=\\square\n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Let ( y=(1 + \frac{6}{x})^{x} )

Take the natural logarithm of both sides: ( \ln y=x\ln(1+\frac{6}{x}) )

Step2: Differentiate both sides with respect to ( x )

Using the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u = x) and (v=\ln(1+\frac{6}{x}))

  • For (u = x), (u^\prime=1)
  • For (v=\ln(1+\frac{6}{x})), use the chain rule. Let (u = 1+\frac{6}{x}), then (v=\ln u). (\frac{dv}{du}=\frac{1}{u}) and (\frac{du}{dx}=-\frac{6}{x^{2}}). So (\frac{dv}{dx}=\frac{-\frac{6}{x^{2}}}{1 + \frac{6}{x}}=\frac{-6}{x(x + 6)})

By the product rule: (\frac{1}{y}y^\prime=\ln(1+\frac{6}{x})+x\times\frac{-6}{x(x + 6)}=\ln(1+\frac{6}{x})-\frac{6}{x + 6})

Step3: Solve for ( y^\prime )

Multiply both sides by ( y=(1+\frac{6}{x})^{x})

(y^\prime=(1+\frac{6}{x})^{x}\left(\ln(1+\frac{6}{x})-\frac{6}{x + 6}\right))

Answer:

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