Consider f(x), g(x) and h(x), for x∈\(\mathbb{R}\) where h(x) = \(\left( {f \circ g} \right)\)(x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

recognizing the need to find h′      (M1)

recognizing the need to find h′ (3) (seen anywhere)      (M1)

evidence of choosing chain rule        (M1)

eg   \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,f'\left( {g\left( 3 \right)} \right) \times g'\left( 3 \right),\,\,f'\left( g \right) \times g'\)

correct working       (A1)

eg  \(f'\left( 7 \right) \times 4,\,\, - 5 \times 4\)

\(h'\left( 3 \right) =  - 20\)      (A1)

evidence of taking their negative reciprocal for normal       (M1)

eg  \( - \frac{1}{{h'\left( 3 \right)}},\,\,{m_1}{m_2} =  - 1\)

gradient of normal is \(\frac{1}{{20}}\)      A1 N4

[7 marks]