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]