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Algebra
Question

(a) Differentiate $$y = ( x ^ { 2 } + 1 ) ^ { 5 }$$ , using the chain rule. Then use the product rule to differentiate $$y = x ( x ^ { 2 } + 1 ) ^ { 5 }$$ .

(b) Differentiate $$y = ( 1 - x ^ { 2 } ) ^ { 4 }$$ . Hence differentiate $$y = 2 x ^ { 3 } ( 1 - x ^ { 2 } ) ^ { 4 }$$ .

(c) Differentiate $$y = ( x ^ { 2 } + x + 1 ) ^ { 3 }$$ , then differentiate $$y = - 2 ( x ^ { 2 } + x + 1 ) ^ { 3 } x$$ .

(d) Similarly, differentiate $$y = x ( 4 - 9 x ^ { 4 } ) ^ { 4 }$$ .

$$\left. \begin{array} { l } { \text { 11(a) } 10 x ( x ^ { 2 } + 1 ) ^ { 4 } , ( x ^ { 2 } + 1 ) ^ { 4 } ( 11 x ^ { 2 } + 1 ) } \\ { \text { (b) } - 8 x ( 1 - x ^ { 2 } ) ^ { 3 } , 2 x ^ { 2 } ( 1 - x ^ { 2 } ) ^ { 3 } ( 3 - 11 x ^ { 2 } ) } \\ { \text { (c) } 3 ( 2 x + 1 ) ( x ^ { 2 } + x + 1 ) ^ { 2 } } \\ { - 2 ( x ^ { 2 } + x + 1 ) ^ { 2 } ( 7 x ^ { 2 } + 4 x + 1 ) } \\ { \text { (d) } ( 4 - 9 x ^ { 4 } ) ^ { 3 } ( 4 - 45 x ^ { 4 } ) } \end{array} \right.$$