Given: f(x) =cos-1\(\cfrac{\text x}2\) + (2x + 7)-1/2
\(\cfrac{d}{d\text x}\)f(x) = cos-1\(\cfrac{\text x}2\) x \(\cfrac{d}{d\text x}\)(2x + 7)-1/2+ \(\left(\cfrac{d}{d\text x}cos^{-1}\cfrac{\text x}2\right)\)(2x + 7)-1/2
\(\left(\because \cfrac{d}{d\text x}u\text v=u\times\cfrac{d\text v}{du}+\cfrac{du}{d\text x}\text v\right)\)
= \(-\cfrac{2}2\)(2x + 7)-3/2cos-1\(\cfrac{\text x}2\) - \(\cfrac{1}{\sqrt{1-\left(\frac{\text x}2\right)^2}}\times\)\(\cfrac12\) (2x + 7)-1/2
(By chain rule)
= -(2x + 7)-3/2 cos-1\(\cfrac{\text x}2\) - \(\cfrac{1}{\sqrt{4-\text x^2}}\) (2x + 7)-1/2
= -(2x + 7)-1/2\(\left(\cfrac{cos^{-1}\cfrac{\text x}2}{2\text x+7}+\cfrac1{\sqrt{4-\text x^2}}\right)\)
= \(-\cfrac{1}{\sqrt{2\text x+ 7}}\) \(\left(\cfrac{cos^{-1}\cfrac{\text x}2}{2\text x+7}+\cfrac1{\sqrt{4-\text x^2}}\right)\)