Standard +0.3 This is a slightly above-average C4 question. Part (i) is a standard proof from first principles using the quotient rule. Part (ii) requires applying the product rule to find dy/dx, evaluating at x=0 (straightforward), then writing the tangent equation. Part (iii) involves solving dy/dx=0 numerically. All techniques are routine for C4, though the combination of exponential and trig functions adds mild complexity.
6. (i) Use the derivative of \(\cos x\) to prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$
The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(ii) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(iii) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).
6. (i) Use the derivative of $\cos x$ to prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$
The curve $C$ has the equation $y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.\\
(ii) Find an equation for the tangent to $C$ at the point where it crosses the $y$-axis.\\
(iii) Find, to 2 decimal places, the $x$-coordinate of the stationary point of $C$.\\
\hfill \mbox{\textit{OCR C4 Q6 [11]}}