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]