Question

$\dfrac{d}{{dx}}\log \left( {\sqrt {x - a} + \sqrt {x - b} } \right) = $
1. $\dfrac{1}{2}\left( {\sqrt {x - a} + \sqrt {x - b} } \right)$
2. $\dfrac{1}{{2\sqrt {\left( {\left( {x - a} \right)\left( {x - b} \right)} \right)} }}$
3. $\dfrac{1}{{\sqrt {\left( {\left( {x - a} \right)\left( {x - b} \right)} \right)} }}$
4. none of these

Answer 1

Hint: In this question, we are given a function $\log \left( {\sqrt {x - a} + \sqrt {x - b} } \right)$and we have to find the first derivative of given function with respect to $x$. Using the differentiation formula $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$ start the solution then use $\dfrac{d}{{dx}}\sqrt x = \dfrac{1}{{2\sqrt x }}$ and solve further terms by taking L.C.M.

Formula Used:
Chain rule –
The chain rule is a formula in calculus that expresses the derivative of the composition of two differentiable functions f and g in terms of f and g's derivatives.
$\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right) \times \dfrac{d}{{dx}}g\left( x \right)$

Complete step by step Solution:
Given that,
$\log \left( {\sqrt {x - a} + \sqrt {x - b} } \right) - - - - - (1)$
Differentiate equation (1) with respect to $x$,
$\dfrac{d}{{dx}}\log \left( {\sqrt {x - a} + \sqrt {x - b} } \right) = \dfrac{1}{{\left( {\sqrt {x - a} + \sqrt {x - b} } \right)}} \times \dfrac{d}{{dx}}\left( {\sqrt {x - a} + \sqrt {x - b} } \right)$
$ = \dfrac{1}{{\left( {\sqrt {x - a} + \sqrt {x - b} } \right)}} \times \left[ {\dfrac{1}{{2\sqrt {x - a} }} + \dfrac{1}{{2\sqrt {x - b} }}} \right]$
Taking $2$ common outside and will solve further by finding L.C.M
L.C.M of $\sqrt {x - a} $and $\sqrt {x - b} $is $\left( {\sqrt {x - a} \sqrt {x - b} } \right)$
$ = \dfrac{1}{{2\left( {\sqrt {x - a} + \sqrt {x - b} } \right)}} \times \left[ {\dfrac{{\sqrt {x - b} + \sqrt {x - a} }}{{\sqrt {x - a} \sqrt {x - b} }}} \right]$
Cancel the like terms from the numerator and denominator
$ = \dfrac{1}{{2\left( {\sqrt {\left( {\left( {x - a} \right)\left( {x - a} \right)} \right)} } \right)}}$

Hence, the correct option is 2.

Note: To solve such a question one should have a good command of the differentiation of logarithm, trigonometry, algebra, and constants. Also, if the function is of double function always apply the chain rule. Using the chain rule, we first find the derivative of an outside function and then the derivative of the inside function.