Write down

(i)     $f'(x)$ ;

(ii)    $g'(x)$ .

Let $h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)$ . Find the exact value of $h'\left( {\frac{\pi }{3}} \right)$ .

(i) $- 3{{\rm{e}}^{ - 3x}}$     A1     N1

(ii) $\cos \left( {x - \frac{\pi }{3}} \right)$     A1     N1

[4 marks]

evidence of choosing product rule     (M1)

e.g. $uv' + vu'$

correct expression     A1

e.g. $- 3{{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right) + {{\rm{e}}^{ - 3x}}\cos \left( {x - \frac{\pi }{3}} \right)$

complete correct substitution of $x = \frac{\pi }{3}$     (A1)

e.g. $- 3{{\rm{e}}^{ - 3\frac{\pi }{3}}}\sin \left( {\frac{\pi }{3} - \frac{\pi }{3}} \right) + {{\rm{e}}^{ - 3\frac{\pi }{3}}}\cos \left( {\frac{\pi }{3} - \frac{\pi }{3}} \right)$        

$h'\left( {\frac{\pi }{3}} \right) = {{\rm{e}}^{ - \pi }}$     A1     N3

[4 marks]

A good number of candidates found the correct derivative expressions in (a). Many applied the product rule, although with mixed success.

Often the substitution of ${\frac{\pi }{3}}$ was incomplete or not done at all.