If f(x)=cos ^{-1}left[frac{1}{sqrt{13}}(2 cos | vedclass

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(B) Given,$f(x)=\cos ^{-1}\left\{\frac{2}{\sqrt{13}} \cos x-\frac{3}{\sqrt{13}} \sin x\right\}$.Let $\cos \alpha = \frac{2}{\sqrt{13}}$ and $\sin \alp

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(B) Given,$f(x)=\cos ^{-1}\left\{\frac{2}{\sqrt{13}} \cos x-\frac{3}{\sqrt{13}} \sin x\right\}$.Let $\cos \alpha = \frac{2}{\sqrt{13}}$ and $\sin \alpha = \frac{3}{\sqrt{13}}$.Then,$f(x)=\cos ^{-1}(\cos \alpha \cos x - \sin \alpha \sin x)$.Using the trigonometric identity $\cos(A+B) = \cos A \cos B - \sin A \sin B$,we get:$f(x) = \cos ^{-1}(\cos(x+\alpha)) = x+\alpha$.Now,differentiating with respect to $x$:$f^{\prime}(x) = \frac{d}{dx}(x+\alpha) = 1 + 0 = 1$.Therefore,$f^{\prime}(0.5) = 1$.

If $f(x)=\cos ^{-1}\left[\frac{1}{\sqrt{13}}(2 \cos x-3 \sin x)\right]$,then $f^{\prime}(0.5)$ is equal to

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