Calc one logarithmic differentiation problem

14 viewed last edited 1 year ago
Timothy Lee
Hey everyone i have a problem i cant seem to figure out and would really apreciate some help the problem is "differentiate y=x^(ln 5x)" i keep getting (5/6x+ln5x+lnx)x^(ln 5x), but the online calculator i have been using is giving me a different answer any help would be greatly apreciated
Given y = x^{(\ln 5x)} Apply the "ln" on both the sides \ln y = \ln x^{(\ln 5x)} \ln y = \ln (5x) * \ln x Differentiate with respect to x \frac{1}{ y} = \frac{1}{x} \ln (5x) + \frac{1}{5x} *5 *\ln x y' = y (\frac{\ln (5x)}{x} + \frac{5*\ln x}{5x}) y' = x^{\ln 5x} (\frac{\ln (5x)}{x} + \frac{5*\ln x}{5x}) y' = x^{\ln 5x} \frac{1}{x}(\ln (5x) + \ln x) y' = x^{\ln 5x - 1}*(\ln (5x) + \ln x)