Let \(f(x) = {{\text{e}}^{2x}}\). The line \(L\) is the tangent to the curve of \(f\) at \((1,{\text{ }}{{\text{e}}^2})\).

Find the equation of \(L\) in the form \(y = ax + b\).

recognising need to differentiate (seen anywhere)     R1

eg     \(f',{\text{ }}2{{\text{e}}^{2x}}\)

attempt to find the gradient when \(x = 1\)     (M1)

eg     \(f'(1)\)

\(f'(1) = 2{{\text{e}}^2}\)     (A1)

attempt to substitute coordinates (in any order) into equation of a straight line     (M1)

eg     \(y - {{\text{e}}^2} = 2{{\text{e}}^2}(x - 1),{\text{ }}{{\text{e}}^2} = 2{{\text{e}}^2}(1) + b\)

correct working     (A1) 

eg     \(y - {{\text{e}}^2} = 2{{\text{e}}^2}x - 2{{\text{e}}^2},{\text{ }}b =  - {{\text{e}}^2}\)

\(y = 2{{\text{e}}^2}x - {{\text{e}}^2}\)     A1     N3

[6 marks]