Question

1)y=(a^2)/2*ln(x+sqrt(x^2-a^2)-(b^2)/2
2)y=sqrt(x)*(e^x-1)*ln*x

Answer 1

$1); ; y=frac{a^2}{2}cdot ln(x+sqrt{x^2-a^2})-frac{b^2}{2}\\(lnu)'=frac{1}{u}cdot u'; ; ,; ; u=x+sqrt{x^2-a^2}\\y'=frac{a^2}{2}cdot frac{1}{x+sqrt{x^2-a^2}}cdot Big (1+frac{2x}{2sqrt{x^2-a^2}}Big )=frac{a^2}{2}cdot frac{1}{x+sqrt{x^2-a^2}}cdot frac{sqrt{x^2-a^2}+x}{sqrt{x^2-a^2}}=\\=frac{a^2}{2cdot sqrt{x^2-a^2}}$

$2); ; y=sqrt{x}cdot (e^{x}-1)cdot lnx\\(uvw)'=u'vw+uv'w+uvw'\\y'=frac{1}{2sqrt{x}}cdot (e^{x}-1)cdot lnx+sqrt{x}cdot e^{x}cdot lnx+sqrt{x}cdot (e^{x}-1)cdot frac{1}{x}$