Derive tan(x)^sin(x)

Question

anonymous · Answer

My closest guess would be something like cosx(tanx)^sinx+sec^2x^sinx, I think...

anonymous · Answer

or $$\cos x(tanx)^{sinx}+\sec ^{2}x ^{sinx}$$

anonymous · Answer

@Callisto hey, sorry to bother you again, but could you help me out on this one?

callisto · Answer

Do you mean to find the derivative of (i) $(\tan x)^{\sin x}$ or  (ii) $\tan (x^{\sin x})$?

anonymous · Answer

i) (tanx)^(sinx)

callisto · Answer

Let's try this:
Let y =  (tanx)^(sinx)
ln y = ln [(tanx)^(sinx)]
ln y = (sinx) [ln (tanx)] 
Differentiate both sides with respect to x:
$$\frac{1}{y}\frac{dy}{dx} = \sin x \frac{d}{dx}[\ln (\tan x)]  + [\ln (\tan x)]  \frac{d}{dx}(\sin x)$$Can you do it from here?

anonymous · Answer

yes, i actually just did that after about half an hour of pondering it. thanks again for the help!

callisto · Answer

Welcome :)