Single Variable Calculus Questions and Answers

Start Your Free Trial

Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 50

Expert Answers info

eNotes eNotes educator | Certified Educator

calendarEducator since 2007

write13,548 answers

starTop subjects are Math, Literature, and Science

Find the derivative of the function $y = (\sin x)^{\ln x}$, using log differentiation

$ \begin{equation} \begin{aligned} \ln y &= \ln (\sin x)^{\ln x}\\ \\ \ln y &= \ln x \ln (\sin x)\\ \\ \frac{d}{dx} \ln y &= \frac{d}{dx} [ \ln x \ln (\sin x)]\\ \\ \frac{1}{y}\frac{dy}{dx} &= \ln x \frac{d}{dx} \ln (\sin x) + \ln(\sin x) \frac{d}{dx} (\ln x)\\ \\ \frac{1}{y} y' &= \ln x \cdot \frac{1}{\sin x} \frac{d}{dx} (\sin x) + \ln (\sin x) \cdot \frac{1}{x}\\ \\ \frac{y'}{y} &= \ln x \cdot \frac{1}{\sin x} \cdot \cos x + \frac{\ln (\sin x)}{x}\\ \\ \frac{y'}{y} &= \ln x \frac{\cos x}{\sin x} + \frac{\ln (\sin x)}{x}\\ \\ \frac{y'}{y} &= \cot x \ln x + \frac{\ln (\sin x)}{x}\\ \\ y' &= y \left( \cot x \ln x + \frac{\ln (\sin x)}{x} \right)\\ \\ y' &= (\sin x)^{\ln x} \left( \cot x \ln x + \frac{\ln (\sin x)}{x} \right) \end{aligned} \end{equation} $