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

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

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