Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.5 Exercises - Page 363: 68

$y=1$

Let $\log x$ denote the natural logarithm $\ln x$. $y=(sinx)^{2x}$ $y’=(sinx)^{2x}(\frac{d}{dx}(\log (sinx)2x)$ $u=\log(sinx)$ $u’=\frac{cos x}{sin x}$ $v=2x$ $v’=2$ $y’=(sinx)^{2x}(2x cotx +2 \log(sin x))$ $y’(\frac{\pi}{2})=2(sin(\frac{\pi}{2}))^{2\frac{\pi}{2}}(\frac{\pi}{2}cot(\frac{\pi}{2})+2\log(sin(\frac{\pi}{2}))$ $=0$ equation of tangent: $y-1=0(x-\frac{\pi}{2})$ $y=1$