(i)     Find the first four derivatives of $f(x)$.

(ii)     Find ${f^{(19)}}(x)$.

(i)     Find the first three derivatives of $g(x)$.

(ii)     Given that ${g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})$, find $p$.

(i)     Find $h'(x)$.

(ii)     Hence, show that $h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}$.

(i)     $f'(x) =  - \sin x,{\text{ }}f''(x) =  - \cos x,{\text{ }}{f^{(3)}}(x) = \sin x,{\text{ }}{f^{(4)}}(x) = \cos x$     A2     N2

(ii)     valid approach     (M1)

eg$\,\,\,\,\,$recognizing that 19 is one less than a multiple of 4, ${f^{(19)}}(x) = {f^{(3)}}(x)$

${f^{(19)}}(x) = \sin x$     A1     N2

[4 marks]

(i)     $g'(x) = k{x^{k - 1}}$

$g''(x) = k(k - 1){x^{k - 2}},{\text{ }}{g^{(3)}}(x) = k(k - 1)(k - 2){x^{k - 3}}$     A1A1     N2

(ii)     METHOD 1

correct working that leads to the correct answer, involving the correct expression for the 19th derivative     A2

eg$\,\,\,\,\,$$k(k - 1)(k - 2) \ldots (k - 18) \times \frac{{(k - 19)!}}{{(k - 19)!}},{{\text{ }}_k}{P_{19}}$

$p = 19$ (accept $\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}$)     A1     N1

METHOD 2

correct working involving recognizing patterns in coefficients of first three derivatives (may be seen in part (b)(i)) leading to a general rule for 19th coefficient     A2

eg$\,\,\,\,\,$$g'' = 2!\left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right),{\text{ }}k(k - 1)(k - 2) = \frac{{k!}}{{(k - 3)!}},{\text{ }}{g^{(3)}}(x){ = _k}{P_3}({x^{k - 3}})$

${g^{(19)}}(x) = 19!\left( {\begin{array}{*{20}{c}} k \\ {19} \end{array}} \right),{\text{ }}19! \times \frac{{k!}}{{(k - 19)! \times 19!}},{{\text{ }}_k}{P_{19}}$

$p = 19$ (accept $\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}$)     A1     N1

[5 marks]

(i)     valid approach using product rule     (M1)

eg$\,\,\,\,\,$$uv' + vu',{\text{ }}{f^{(19)}}{g^{(20)}} + {f^{(20)}}{g^{(19)}}$

correct 20th derivatives (must be seen in product rule)     (A1)(A1)

eg$\,\,\,\,\,$${g^{(20)}}(x) = \frac{{21!}}{{(21 - 20)!}}x,{\text{ }}{f^{(20)}}(x) = \cos x$

$h'(x) = \sin x(21!x) + \cos x\left( {\frac{{21!}}{2}{x^2}} \right){\text{ }}\left( {{\text{accept }}\sin x\left( {\frac{{21!}}{{1!}}x} \right) + \cos x\left( {\frac{{21!}}{{2!}}{x^2}} \right)} \right)$    A1     N3

(ii)     substituting $x = \pi$ (seen anywhere)     (A1)

eg$\,\,\,\,\,$${f^{(19)}}(\pi ){g^{(20)}}(\pi ) + {f^{(20)}}(\pi ){g^{(19)}}(\pi ),{\text{ }}\sin \pi \frac{{21!}}{{1!}}\pi  + \cos \pi \frac{{21!}}{{2!}}{\pi ^2}$

evidence of one correct value for $\sin \pi$ or $\cos \pi$ (seen anywhere)     (A1)

eg$\,\,\,\,\,$$\sin \pi  = 0,{\text{ }}\cos \pi  =  - 1$

evidence of correct values substituted into $h'(\pi )$     A1

eg$\,\,\,\,\,$$21!(\pi )\left( {0 - \frac{\pi }{{2!}}} \right),{\text{ }}21!(\pi )\left( { - \frac{\pi }{2}} \right),{\text{ }}0 + ( - 1)\frac{{21!}}{2}{\pi ^2}$

Note: If candidates write only the first line followed by the answer, award A1A0A0.

$\frac{{ - 21!}}{2}{\pi ^2}$     AG     N0

[7 marks]