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

• A

  $0.5$
• Correct answer
  B

  1
• C

  0
• D

  $-1$

Solution

Given, $f(x)=\cos ^{-1}\left\{\frac{1}{\sqrt{13}}(2 \cos x-3 \sin x)\right\}$ $\Rightarrow f(x)=\cos ^{-1}\left\{\frac{2}{\sqrt{13}} \cos x-\frac{3}{\sqrt{13}} \sin x\right\}$

$\Rightarrow \quad f(x)=\cos ^{-1}\{\cos \alpha \cdot \cos x-\sin \alpha \cdot \sin x\}$ $\begin{aligned} &=\cos ^{-1}\{\cos (x+\alpha)\} \\ &=x+\alpha \end{aligned}$ On differentiating w.r.t. $x$, we get $\begin{aligned} f^{\prime}(x) &=1=\text { constant value } \\ \therefore \quad f^{\prime}(0.5) &=1 \end{aligned}$

Hence, the correct answer is option: 'B'.