Question

# If ( y=log sqrt{x+sqrt{x^{2}+a^{2}}} ) then ( d y / d x ) is

Solution

( y=log sqrt{x+sqrt{x^{2}+a^{2}}} )

( y=frac{1}{2} log (x+sqrt{x^{2}+a^{2}}) Rightarrow 2 y=log (x+sqrt{x^{2}+a^{2}}) )

( frac{a d y}{d x}=left(frac{1}{x+sqrt{x^{2}+a^{2}}}right)left(1+frac{2 x}{2 sqrt{x^{2}+a^{2}}}right) )

( frac{d y}{d x}=left(frac{1}{x+sqrt{x^{2}+a^{2}}}right)left(frac{x+sqrt{x^{2}+a^{2}}}{sqrt{x^{2}+a^{2}}}right) )

( frac{d y}{d x}=frac{1}{2 sqrt{x^{2}+a^{2}}} )