Standard +0.3 This question requires finding dy/dx using the quotient rule with logarithms, then solving 4 = (3+2ln x)/(1+ln x) for x, and finally substituting to find the gradient. It's slightly above average difficulty due to the algebraic manipulation needed and the reverse-lookup step, but remains a standard multi-step calculus problem with familiar techniques.
Use quotient rule (or product rule) to find first derivative
\*M1
Must have correct \(u\) and \(v\)
Obtain \(-\dfrac{1}{x(1+\ln x)^2}\) or (unsimplified) equivalent
A1
Use \(y=4\) to obtain \(\ln x = -\frac{1}{2}\) or exact equivalent for \(x\)
B1
Substitute for \(x\) in their first derivative
DM1
Obtain \(-4e^{\frac{1}{2}}\) or exact equivalent
A1
Must be simplified to contain a single exponential term
Total
5
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use quotient rule (or product rule) to find first derivative | \*M1 | Must have correct $u$ and $v$ |
| Obtain $-\dfrac{1}{x(1+\ln x)^2}$ or (unsimplified) equivalent | A1 | |
| Use $y=4$ to obtain $\ln x = -\frac{1}{2}$ or exact equivalent for $x$ | B1 | |
| Substitute for $x$ in their first derivative | DM1 | |
| Obtain $-4e^{\frac{1}{2}}$ or exact equivalent | A1 | Must be simplified to contain a single exponential term |
| **Total** | **5** | |
3 A curve has equation $y = \frac { 3 + 2 \ln x } { 1 + \ln x }$. Find the exact gradient of the curve at the point for which $y = 4$.\\
\hfill \mbox{\textit{CAIE P2 2019 Q3 [5]}}