A curve has equation $y = e^{-2x} \tan x$, where $0 \le x < \frac{1}{2}\pi$.
(i)[5]
Obtain an expression for $\frac{dy}{dx}$ and demonstrate that it can be expressed in the form $e^{-2x}(a + b \tan x)^2$, where $a$ and $b$ are constants.
(ii)[1]
Explain why the curve's gradient is never negative.
(iii)[1]
Find the value of $x$ at which the gradient is smallest.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply that differentiating $e^{-2x}$ gives $-2e^{-2x}$” …