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

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Solution

Let $y = (sin x - cos x)^{(sin x - cos x)}$
Taking logarithm on both the sides, we obtain
$log y = log [(sin x - cos x)^{(sin x - cos x)}]$
$\Rightarrow log y = (sin x - cos x) . log (sin x - cos x)$
Differentiating both sides with respect to x, we obtain
$\frac{1}{y} \frac{d y}{d x} = \frac{d}{d x} [(sin x - cos x) . log (sin x - cos x)$
$\Rightarrow \frac{1}{y} \frac{d y}{d x} = log (sin x - cos x) . \frac{d}{d x} (sin x - cos x) + (sin x - cos x) . \frac{d}{d x} log (sin x - cos x)$
$\Rightarrow \frac{1}{y} \frac{d y}{d x} = log (sin x - cos x) . (cos x + sin x) + (sin x - cos x) . \frac{1}{(sin x - cos x)} . \frac{d}{d x} (sin x - cos x)$
$\Rightarrow \frac{d y}{d x} = (sin x - cos x)^{sin x - cos x} [(cos x + sin x) . log (sin x - cos x) + (cos x + sin x)]$
$∴ \frac{d y}{d x} = (sin x - cos x)^{(sin x - cos x)} (cos x + sin x) [1 + log (sin x - cos x)]$

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