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.