Question

solve this question pls​

Answer 1

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$\bullet \underline{ \bf{derivative \: of \: log(x + \sqrt{ {x}^{2} + {a}^{2}) } \: wrt \: x}}$

$\: \: \ \: \bigstar \underline{ \displaystyle \sf{given \: equation}}$

$\: \dashrightarrow \bf{log(x + \sqrt{ {x}^{2} + {a}^{2} ) } }$

$\: \bigstar\huge \underline{ \sf{answer}}$

$\: \dashrightarrow \displaystyle \sf{log(x + \sqrt{ {x}^{2} + {a}^{2} )} }$

$\: \underline {\bf{differentiating \: the \: equation \: with \: respect \: to \: x}}$

$\: \dashrightarrow \displaystyle \sf{ log(x + \sqrt{ {x}^{2} + {a}^{2}) } }$

$\dashrightarrow \displaystyle \sf{ \frac{dy}{dx} = log(x + \sqrt{ {x}^{2} + {a}^{2})} dx }$

$\: \dashrightarrow\displaystyle \sf{ \frac{dy}{dx} = \frac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \times 1 + \frac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } \times 2x}$

$\: \dashrightarrow \displaystyle \sf{ \frac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } + \frac{2x}{ \sqrt{ {x}^{2} + {a}^{2} } } }$

$\: \\ \dashrightarrow \displaystyle \sf{ \frac{1}{ x + \sqrt{ {x}^{2} + {a}^{2} } } + \bigg(1 + \frac{1}{2} { ({x}^{2} + a) }^{ \frac{ - 1}{2} } \times 2x \bigg)}$

$\: \\ \dashrightarrow \displaystyle \sf{ \frac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \bigg(1 + \frac{x}{ \sqrt{ {x}^{2} + {a}^{2} } } } \bigg)$

$\\ \dashrightarrow \displaystyle \sf{ \frac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \bigg(1 + \frac{x}{ {x}^{2} + {a}^{2} } \bigg)}$

$\: \\ \dashrightarrow \displaystyle \sf{ \frac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \bigg(1 + \frac{x}{ \sqrt{ {x}^{2} + {a }^{2}} } } \bigg)$

$\: \dashrightarrow \displaystyle \sf{ \frac{ \sqrt{ {x}^{2} + {a}^{2} } + x }{(x + \sqrt{ {x}^{2} + {a}^{2} } \times \sqrt{ {x}^{2} + {a}^{2} } } }$

$\: \dashrightarrow \displaystyle \sf{ \cancel \frac{ \sqrt{ {x}^{2} + {a}^{2} } + x }{x + \sqrt{ {x}^{2} + {a}^{2} } } \times \frac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } }$

$\: \dashrightarrow \displaystyle \sf{ \frac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } }$

$\: \bigstar \boxed {\bf \frac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } is \: the \: answer }$