Exercise:
nprvmulticols abclist abc displaystyle fracddddx left fracx^ax^b + right abc displaystyle fracddddy left fracsinyy^n + right abc displaystyle fracddddz left fracz^m + cosz right abc displaystyle fracddddt left fraca^tt^ + right abc displaystyle fracddddphi left fraclnphiphi^p + right abc displaystyle fracddddtheta left fractheta costhetatheta + sintheta right abc displaystyle fracddddr left fractanrr lnr right abc displaystyle fracddddu left fracsinhuu^ + coshu right abc displaystyle fracddddx left fracx^ lnxx^ + b^x right abc displaystyle fracddddy left fracy^n sin^y + cosy right abc displaystyle fracddddz left fraca^z^lnz^ + right abc displaystyle fracddddt left fractant^c^t right abc displaystyle fracddddphi left fraccosh^phiphi^ + phi + right abc displaystyle fracddddx left fraclog_ax^ + x b^x right abc displaystyle fracddddalpha left fracalpha^a + alpha^blnalpha + alpha right abc displaystyle fracddddr left fracr sinr^cosr right abc displaystyle fracddddtheta left fracsinhtheta^theta^ + right abc displaystyle fracddddz left fracz coszz^ + sinz right abc displaystyle fracddddt left fraca^t lntt + right abc displaystyle fracddddy left fracy^ + cosylog_by^ + right abc displaystyle fracddddchi left fracchi^n c^chisinhchi right abc displaystyle fracddddv left fracv^ + a^cosv right abc displaystyle fracddddq left fractanqlog_ca^q^ right abc displaystyle fracddddx left fracx^ + sinx^lnx^ + right abclist nprvmulticols

Solution:
abclist abc fraca x^a- x^b + - x^a b x^b-x^b + ^ abc fracy^n + cosy - siny n y^n - y^n + ^ abc fracz^ cosz + z^ + sinzcos^z abc fraca^t ln a t^ + - a^t ln a tt^ + ^ abc fracfracphiphi^p + - lnphi p phi^p - phi^p + ^ abc Use quotient rule:  Numerator: costheta - theta sintheta  Denominator: theta + sintheta^  Full derivative: apply product rule in numerator and chain rule in denominator. abc Use quotient rule:  Numerator: sec^r r ln r - tanr ln r +  Denominator: r ln r^ abc Use quotient rule:  Numerator: coshuu + sinhuu^ + coshu - sinhu u + sinhu  Denominator: u^ + coshu^ abc Use quotient and product rule:  Numerator: x ln x + xx^ + b^x - x^ ln x x + b^x ln b  Denominator: x^ + b^x^ abc Apply quotient rule and chain/product rules to numerator. abc fracz a^z^ ln a lnz^ + - a^z^ ln a fraczz^ + lnz^ + ^ abc fract sec^t^ c^t ln c - tant^ c^t ln cc^t ln c^ abc Use product and quotient rule:  Numerator: phi coshphi sinhphi phi^ + phi + - cosh^phi phi +  Denominator: phi^ + phi + ^ abc Use quotient rule:  Numerator: left fracxx^ + fracln a right x b^x + log_ax^ + b^x + x b^x ln b  Denominator: x b^x^ abc Quotient rule:  Numerator: a alpha^a - + b alpha^b - ln alpha + alpha - alpha^a + alpha^bfracalpha +  Denominator: ln alpha + alpha^ abc Apply quotient and chain rule:  Numerator: sinr^ + r r cosr^  Then apply quotient rule with cosr in the denominator. abc fractheta coshtheta^theta^ + - sinhtheta^ thetatheta^ + ^ abc Use product rule in numerator and chain rule in denominator. abc Use quotient rule:  Numerator: a^t ln a lnt + a^t fract  Denominator: t + ^ abc Apply quotient rule to:  Numerator: y + -siny  Denominator: left log_by^ + right^ fracyy^ + ln b abc Use quotient rule:  Numerator: n chi^n - c^chi ln c + chi^n c^chi ln c  Denominator: sinhchi  Then: minus chi^n c^chi coshchi  Denominator squared. abc Use quotient rule:  Numerator: v a^cosv - v^ + a^cosv -sinv ln a  Denominator: a^cosv abc fracsec^q log_ca^q^ - tanq fracq ln aln clog_ca^q^^ abc Apply product and chain rule to numerator:  Numerator: x^ + sinxx + cosx  Denominator: lnx^ + ^ with derivative fracxx^ + abclist