Chapter 2 - The Derivative - 2.4 The Product And Quotient Rules - Exercises Set 2.4 - Page 147: 18

$$\eqalign{ & \left( {\text{a}} \right)f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr & \left( {\text{b}} \right)f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr} $$

$$\eqalign{ & f\left( x \right) = {x^{ - 5}}\left( {{x^2} + 2x} \right)\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right) \cr & \cr & \left( {\text{a}} \right){\text{By multiplying}} \cr & f\left( x \right) = \left( {{x^{ - 3}} + 2{x^{ - 4}}} \right)\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right) \cr & f\left( x \right) = \left( {8{x^{ - 4}} - 2{x^{ - 3}} - 3{x^{ - 2}}} \right)\left( {2{x^9} + 1} \right) \cr & f\left( x \right) = - 6{x^7} - 4{x^6} + 16{x^5} + 8{x^{ - 4}} - 2{x^{ - 3}} - 3{x^{ - 2}} \cr & {\text{Differentiating}} \cr & f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr & \cr & \left( {\text{b}} \right){\text{By using the product rule}} \cr & f\left( x \right) = \left[ {{x^{ - 5}}\left( {{x^2} + 2x} \right)} \right]\left[ {\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right)} \right] \cr & f'\left( x \right) = \left[ {{x^{ - 5}}\left( {{x^2} + 2x} \right)} \right]\frac{d}{{dx}}\left[ {\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left[ {\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right)} \right]\frac{d}{{dx}}\left[ {{x^{ - 5}}\left( {{x^2} + 2x} \right)} \right] \cr & \cr & f'\left( x \right) = \left[ {{x^{ - 5}}\left( {{x^2} + 2x} \right)} \right]\left[ {\left( {4 - 3x} \right)\frac{d}{{dx}}\left[ {2{x^9} + 1} \right] + \left( {2{x^9} + 1} \right)\frac{d}{{dx}}\left( {4 - 3x} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left[ {\left( {4 - 3x} \right)\left( {2{x^9} + 1} \right)} \right]\left[ {{x^{ - 5}}\frac{d}{{dx}}\left[ {{x^2} + 2x} \right] + \left( {{x^2} + 2x} \right)\frac{d}{{dx}}\left[ {{x^{ - 5}}} \right]} \right] \cr & \cr & f'\left( x \right) = \left( {{x^{ - 3}} + 3{x^{ - 4}}} \right)\left[ {\left( {4 - 3x} \right)\left( {18{x^8}} \right) + \left( {2{x^9} + 1} \right)\left( { - 3} \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 6{x^{10}} + 8{x^9} - 3x + 4} \right)\left[ {{x^{ - 5}}\left( {2x + 2} \right) + \left( {{x^2} + 2x} \right)\left( { - 5{x^{ - 6}}} \right)} \right] \cr & {\text{Simplify }} \cr & f'\left( x \right) = \left( {{x^{ - 3}} + 3{x^{ - 4}}} \right)\left( {72{x^8} - 54{x^9} - 6{x^9} - 3} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 6{x^{10}} + 8{x^9} - 3x + 4} \right)\left( {2{x^{ - 4}} + 2{x^{ - 5}} - 10{x^{ - 5}} - 5{x^{ - 4}}} \right) \cr & f'\left( x \right) = \left( {{x^{ - 3}} + 3{x^{ - 4}}} \right)\left( {72{x^8} - 60{x^9} - 3} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 6{x^{10}} + 8{x^9} - 3x + 4} \right)\left( { - 3{x^{ - 4}} - 8{x^{ - 5}}} \right) \cr & f'\left( x \right) = \left( { - 9{x^{ - 4}} - 3{x^{ - 3}} + 216{x^4} - 108{x^5} - 60{x^6}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 32{x^{ - 5}} + 12{x^{ - 4}} + 9{x^{ - 3}} - 64{x^4} + 24{x^5} + 18{x^6}} \right) \cr & {\text{Combining like terms}} \cr & f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr & \cr & \left( {\text{a}} \right)f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr & \left( {\text{b}} \right)f\left( x \right) = - 42{x^6} - 24{x^5} + 80{x^4} - 32{x^{ - 5}} + 6{x^{ - 4}} + 6{x^{ - 3}} \cr} $$