$f(x)=\dfrac{\sin^3x+\cos^3x}{1-\sin x.\cos x}\\\hspace{0,9cm}=\dfrac{(\sin x + \cos x)(\sin^2x-\sin x.\cos x+\cos^2x)}{1-\sin x.cos x}\\\hspace{0,9cm}=\dfrac{(\sin x + \cos x)(1-\sin x.\cos x)}{1-\sin x.cos x}\\\hspace{0,9cm}=\sin x + \cos x\\\to f\Bigg(\dfrac{\pi}{4}\Bigg)=\sin\dfrac{\pi}{4}+\cos \dfrac{\pi}{4}=\sqrt{2}\\f(x)=\sin x + \cos x \to f'(x)=\cos x - \sin x \\\to f'\Bigg(\dfrac{\pi}{4}\Bigg)=\cos \dfrac{\pi}{4}-\sin \dfrac{\pi}{4}=0\\\to f\Bigg(\dfrac{\pi}{4}\Bigg)-\sqrt{2}f'\Bigg(\dfrac{\pi}{4}\Bigg)=\sqrt{2}$
