\(\displaystyle a) \ f(x) \ = \ ln\bigg[\frac{|1-x|^{1/2}}{(x^{2}+2)^{3}}\bigg] \ = \ ln|1-x|^{1/2}-ln(x^{2}+2)^{3}\)
\(\displaystyle = \ \frac{1}{2}ln|1-x|-3ln(x^{2}+2)\)
\(\displaystyle Hence, \ f'(x) \ = \ \frac{1}{2}\bigg(\frac{-1}{1-x}\bigg)-3\bigg(\frac{2x}{x^{2}+2}\bigg) \ = \ \frac{1}{2x-2}-\frac{6x}{x^{2}+2}\)
\(\displaystyle c) \ y \ = \ x^{sin(x)}\)
\(\displaystyle ln|y| \ = \ ln|x^{sin(x)}|, \ take \ log \ of \ both \ sides\)
\(\displaystyle ln|y| \ = \ sin(x)ln|x|, \ property \ of \ logs\)
\(\displaystyle \frac{y'}{y} \ = \ cos(x)ln|x|+sin(x)/x, \ implicit \ differentation\)
\(\displaystyle y' \ = \ y[cos(x)ln|x|+sin(x)/x]\)
\(\displaystyle y' \ = \ x^{sin(x)}[cos(x)ln|x|+sin(x)/x], \ resub.\)
\(\displaystyle Note: \ The \ sticking \ point \ here \ is \ one \ should \ have \ a \ good \ grasp \ of \ log \ properties.\)
