Standard +0.3 This is a straightforward application of the product rule combined with chain rule for standard trigonometric functions. While it requires careful differentiation of tan(x/2) and cos(2x), the process is mechanical with no problem-solving insight needed. The evaluation at a specific point is routine substitution, making this slightly easier than average.
2 A curve has equation \(y = 3 \tan \frac { 1 } { 2 } x \cos 2 x\).
Find the gradient of the curve at the point for which \(x = \frac { 1 } { 3 } \pi\).
Substitute \(\frac{1}{3}\pi\) into attempt at first derivative and attempt evaluation to find the gradient
DM1
Obtain \(-4\)
A1
Total
5
**Question 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply derivative of $\tan\frac{1}{2}x$ is $\frac{1}{2}\sec^2\frac{1}{2}x$ or derivative of $\cos2x$ is $-2\sin2x$ | B1 | |
| Attempt use of product rule to find first derivative | \*M1 | |
| Obtain correct $\frac{3}{2}\sec^2\frac{1}{2}x\cos2x - 6\tan\frac{1}{2}x\sin2x$ | A1 | or (unsimplified) equivalent |
| Substitute $\frac{1}{3}\pi$ into attempt at first derivative and attempt evaluation to find the gradient | DM1 | |
| Obtain $-4$ | A1 | |
| **Total** | **5** | |
2 A curve has equation $y = 3 \tan \frac { 1 } { 2 } x \cos 2 x$.\\
Find the gradient of the curve at the point for which $x = \frac { 1 } { 3 } \pi$.\\
\hfill \mbox{\textit{CAIE P2 2023 Q2 [5]}}