Continuity and Differentiability — Class 12 Maths Solution

Let $y = \sqrt {\cfrac{{(x - 1)(x - 2)}}{{(x - 3)(x - 4)(x - 5)}}}$ ….(i)

By taking log on both sides of (i), we get

$\log y = \cfrac{1}{2}[\log (x - 1) + \log (x - 2)$ $- \log (x - 3) - \log (x - 4) - \log (x - 5)$] …(ii)

Now, differentiating (ii) on both sides w.r.t, x, we get

$\cfrac{1}{y} \cdot \cfrac{{dy}}{{dx}} = \cfrac{1}{2}\left[ {\cfrac{1}{{(x - 1)}} + \cfrac{1}{{(x - 2)}} - \cfrac{1}{{(x - 3)}} - \cfrac{1}{{(x - 4)}} - \cfrac{1}{{(x - 5)}}} \right]$

therefore, $\cfrac{{dy}}{{dx}} = \cfrac{1}{2}\sqrt {\cfrac{{(x - 1)(x - 2)}}{{(x - 3)(x - 4)(x - 5)}}} \times$ $\left[ {\cfrac{1}{{(x - 1)}} + \cfrac{1}{{(x - 2)}} - \cfrac{1}{{(x - 3)}} - \cfrac{1}{{(x - 4)}} - \cfrac{1}{{(x - 5)}}} \right]$