Answer
$$\eqalign{
& \left( {\text{a}} \right)f'\left( x \right) = 3\left( {7{x^6} + 2} \right){\left( {{x^7} + 2x - 3} \right)^2} \cr
& \left( {\text{b}} \right)f'\left( x \right) = 3\left( {7{x^6} + 2} \right){\left( {{x^7} + 2x - 3} \right)^2} \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\left( {{x^7} + 2x - 3} \right)^3} \cr
& \left( {\text{a}} \right){\text{By multiplying use }}{\left( {a + b + c} \right)^3} \cr
& = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} + 3{a^2}c + 6abc + 3{b^2}c + 3a{c^2} + 3b{c^2} + {c^3} \cr
& {\text{We obtain}} \cr
& f\left( x \right) = {x^{21}} + 6{x^{15}} - 9{x^{14}} + 12{x^9} - 36{x^8} + 27{x^7} + 8{x^3} - 36{x^2} + 54x - 27 \cr
& {\text{Differentiating}} \cr
& f'\left( x \right) = 21{x^{20}} + 90{x^{14}} - 126{x^{13}} - 288{x^7} + 189{x^6} + 24{x^2} - 72x + 54 \cr
& {\text{Factoring we obtain}} \cr
& f'\left( x \right) = 3\left( {7{x^6} + 2} \right){\left( {{x^7} + 2x - 3} \right)^2} \cr
& \cr
& \left( {\text{b}} \right){\text{By using the product rule}} \cr
& f\left( x \right) = \left( {{x^7} + 2x - 3} \right)\left( {{x^7} + 2x - 3} \right)\left( {{x^7} + 2x - 3} \right) \cr
& f'\left( x \right) = \left( {{x^7} + 2x - 3} \right)\frac{d}{{dx}}\left[ {\left( {{x^7} + 2x - 3} \right)\left( {{x^7} + 2x - 3} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left( {{x^7} + 2x - 3} \right)^2}\frac{d}{{dx}}\left[ {{x^7} + 2x - 3} \right] \cr
& f'\left( x \right) = \left( {{x^7} + 2x - 3} \right)\left[ {2\left( {{x^7} + 2x - 3} \right)\frac{d}{{dx}}\left( {{x^7} + 2x - 3} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left( {{x^7} + 2x - 3} \right)^2}\frac{d}{{dx}}\left[ {{x^7} + 2x - 3} \right] \cr
& {\text{Computing derivatives}} \cr
& f'\left( x \right) = \left( {{x^7} + 2x - 3} \right)\left[ {2\left( {{x^7} + 2x - 3} \right)\left( {7{x^6} + 2} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left( {{x^7} + 2x - 3} \right)^2}\left( {7{x^6} + 2} \right) \cr
& f'\left( x \right) = 2{\left( {{x^7} + 2x - 3} \right)^2}\left( {7{x^6} + 2} \right) + {\left( {{x^7} + 2x - 3} \right)^2}\left( {7{x^6} + 2} \right) \cr
& {\text{Simplify }} \cr
& f'\left( x \right) = 3\left( {7{x^6} + 2} \right){\left( {{x^7} + 2x - 3} \right)^2} \cr} $$
