First slide

Derivatives

Question

$If y = \sqrt{\frac{(x - a) (x - b)}{(x - c) (x - d)}} then \frac{d y}{d x} =$

Moderate

A

$\frac{y}{2} [\frac{1}{x - a} + \frac{1}{x - b} - \frac{1}{x - c} - \frac{1}{x - d}]$
B

$y [\frac{1}{x - a} + \frac{1}{x - b} - \frac{1}{x - c} - \frac{1}{x - d}]$
C

$\frac{1}{2} [\frac{1}{x - a} + \frac{1}{x - b} - \frac{1}{x - c} - \frac{1}{x - d}]$
D

None of these

Solution

$\log y = \frac{1}{2} [\log (x - a) + \log (x - b) - \log (x - c) - \log (x - d)]$
$\begin{aligned} \frac{1}{y} \frac{d y}{d x} = \frac{1}{2} [\frac{1}{x - a} + \frac{1}{x - b} - \frac{1}{x - c} - \frac{1}{x - d}] \\ \frac{d y}{d x} = \frac{y}{2} [\frac{1}{x - a} + \frac{1}{x - b} - \frac{1}{x - c} - \frac{1}{x - d}] \end{aligned}$

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