Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative, \(f’(x)\).

METHOD 1 

derivative of \(f(x)\)     A2

\(7{({x^2} + 3)^6}(x2)\)

recognizing need to find \({x^4}\) term in \({({x^2} + 3)^6}\) (seen anywhere)     R1

eg\(\,\,\,\,\,\)\(14x{\text{ (term in }}{x^4})\)

valid approach to find the terms in \({({x^2} + 3)^6}\)     (M1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ r \end{array}} \right){({x^2})^{6 - r}}{(3)^r},{\text{ }}{({x^2})^6}{(3)^0} + {({x^2})^5}{(3)^1} +  \ldots \), Pascal’s triangle to 6th row

identifying correct term (may be indicated in expansion)     (A1)

eg\(\,\,\,\,\,\)\({\text{5th term, }}r = 2,{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right),{\text{ }}{({x^2})^2}{(3)^4}\)

correct working (may be seen in expansion)     (A1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right){({x^2})^2}{(3)^4},{\text{ }}15 \times {3^4},{\text{ }}14x \times 15 \times 81{({x^2})^2}\)

\(17010{x^5}\)     A1     N3

METHOD 2

recognition of need to find \({x^6}\) in \({({x^2} + 3)^7}\) (seen anywhere) R1 

valid approach to find the terms in \({({x^2} + 3)^7}\)     (M1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ r \end{array}} \right){({x^2})^{7 - r}}{(3)^r},{\text{ }}{({x^2})^7}{(3)^0} + {({x^2})^6}{(3)^1} +  \ldots \), Pascal’s triangle to 7th row

identifying correct term (may be indicated in expansion)     (A1)

eg\(\,\,\,\,\,\)6th term, \(r = 3,{\text{ }}\left( {\begin{array}{*{20}{c}} 7 \\ 3 \end{array}} \right),{\text{ (}}{{\text{x}}^2}{)^3}{(3)^4}\)

correct working (may be seen in expansion)     (A1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 4 \end{array}} \right){{\text{(}}{{\text{x}}^2})^3}{(3)^4},{\text{ }}35 \times {3^4}\)

correct term     (A1)

\(2835{x^6}\)

differentiating their term in \({x^6}\)     (M1)

eg\(\,\,\,\,\,\)\((2835{x^6})',{\text{ (6)(2835}}{{\text{x}}^5})\)

\(17010{x^5}\)     A1     N3

[7 marks]