Question

Find dericative of (lnx)^cosx

Answer 1

General rule of thumb:
\[\begin{align*}\frac{d}{dx}\left[f(x)^{g(x)}\right]&=\frac{d}{dx}\left[\exp\left(g(x)\ln f(x)\right)\right]\\[1ex]
&=\frac{d}{dx}\left[g(x)\ln f(x)\right]\times\exp\left(g(x)\ln f(x)\right)\\[1ex]
&=\left(\frac{d}{dx}[g(x)]\ln f(x)+\frac{g(x)\frac{d}{dx}[f(x)]}{f(x)}\right)\times\exp\left(g(x)\ln f(x)\right)\\[1ex]
&=\left(\frac{d}{dx}[g(x)]\ln f(x)+\frac{g(x)\frac{d}{dx}[f(x)]}{f(x)}\right)\times f(x)^{g(x)}\end{align*}\]

Answer 2

(1) \(a^b=\exp\left(\ln a^b\right)=\exp(b\ln a)\)

(2) chain rule

(3) product rule

(4) see (1)

Answer 3

Its hard to see when do you use the product rule I think I missed that part?