Question

Differentiate w.r.t. x the function:$(sinx-cosx)^{(sinx-cosx)},\frac{π}{4}<x<\frac{3π}{4}$

Answer 1

The correct answer is $(sinx-cosx)^{(sinx-cosx)}(cosx+sinx)[1+log(sinx-cosx)]$
Let $y=(sinx-cosx)^{(sinx-cosx)}$
Taking logarithm on both the sides,we obtain
$log\,y=log[(sinx-cosx)^{(sinx-cosx)}]$
$⇒log\,y=(sin\,x-cos\,x).log(sin\,x-cos\,x)$
Differentiating both sides with respect to $x$, we obtain
$\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}[(sin\,x-cos\,x)log(sinx-cosx)]$
$⇒\frac{1}{y}\frac{dy}{dx}=log(sinx-cosx).\frac{d}{dx}(sinx-cosx)+(sinx-cosx).\frac{d}{dx}log(sinx-cosx)$
$⇒\frac{1}{y}\frac{dy}{dx}=log(sinx-cosx).(cosx+sinx)+(sinx-cosx).\frac{1}{(sinx-cosx)}.\frac{d}{dx}(sinx-cosx)$
$⇒\frac{1}{y}\frac{dy}{dx}=(sinx-cosx)^{(sinx-cosx)}[(cosx+sinx).log(sinx-cosx)+(cosx+sinx)]$
$∴\frac{dy}{dx}=(sinx-cosx)^{(sinx-cosx)}(cosx+sinx)[1+log(sinx-cosx)]$