[tex]a)( x^{2} *e^x)'=( x^{2} )'*e^x+ x^{2} *(e^x)'=2xe^x+ x^{2} e^x=xe^x(2+x) \\ b)[( x^{2} +x+1)lnx]'=(2x+1)lnx+ \frac{x^2+x+1}{x} \\ c)(2^xlnx+e^x log_{2}x)'=2^xln^2x+ \frac{2^x}{x}+e^x log_{2}x+ \frac{e^x}{xln2} \\ f)(e^xsinx)'=e^xsinx-e^xcosx=e^x(sinx-cosx) \\ d)[(x+1)* \sqrt{x} ]'= \sqrt{x} + \frac{x+1}{2 \sqrt{x} } \\ e)[( x^{2} +x+7)* \sqrt[3]{x} ]'=(2x+1) \sqrt[3]{x}+ \frac{ x^{2} +x+7}{3 \sqrt{x} } \\ g)[( x^{3}-x)tgx]' =(3 x^{2} -1)tgx+ \frac{x^3-x}{cos^2x} [/tex]
[tex]h)(xarcsinx)'=arcsinx+ \frac{x}{ \sqrt{1-x^2} } \\ i)(arcsinx*arccosx)'= \frac{arccosx}{ \sqrt{1-x^2} } - \frac{arcsinx}{ \sqrt{1-x^2} } \\ h)(arctgx*arcctgx)'= \frac{arcctgx}{1+ x^{2} } - \frac{arctgx}{1+ x^{2} } \\ k)( x^{2} lnx+tgxsinx)'=2xlnx+x+ \frac{sinx}{cos^2x}+cosxtgx[/tex]
