| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2020 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative satisfies condition |
| Difficulty | Standard +0.8 This question requires logarithmic differentiation of an exponential function with variable base and exponent, followed by algebraic manipulation to derive a transcendental equation. While the technique is standard for P4/Further Maths, it demands careful application of implicit differentiation, product rule, and algebraic rearrangement across multiple steps—more challenging than routine differentiation but less demanding than proof-based or multi-concept integration problems. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = x^{\sin x} \Rightarrow \ln y = \sin x \ln x\) | B1 | o.e.; condone \(\log y = \sin x \log x\) |
| \(\frac{1}{y}\frac{dy}{dx} = \frac{\sin x}{x} + \ln x\cos x\) | M1 M1 A1 | M1: \(\ln y \to \frac{1}{y}\times\frac{dy}{dx}\); M1: product rule on \(\sin x \ln x\) (must be correct if quoted); A1: correct differentiation |
| \(\frac{dy}{dx} = \frac{y\sin x}{x} + y\ln x\cos x\) | A1 | o.e.; accept in terms of \(x\) only; ISW after correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(\frac{dy}{dx}=0\), divides by \(y\) or \(x^{\sin x}\) | M1 | \(\frac{dy}{dx}\) must be in correct form |
| \(\frac{\sin x}{\cos x} + x\ln x = 0 \Rightarrow \tan x + x\ln x = 0\) | A1* | Must show intermediate line \(\frac{\sin x}{\cos x}+x\ln x=0\); can be implied by \(\sin x + x\ln x\cos x = 0 \div\cos x\) |
# Question 6:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = x^{\sin x} \Rightarrow \ln y = \sin x \ln x$ | B1 | o.e.; condone $\log y = \sin x \log x$ |
| $\frac{1}{y}\frac{dy}{dx} = \frac{\sin x}{x} + \ln x\cos x$ | M1 M1 A1 | M1: $\ln y \to \frac{1}{y}\times\frac{dy}{dx}$; M1: product rule on $\sin x \ln x$ (must be correct if quoted); A1: correct differentiation |
| $\frac{dy}{dx} = \frac{y\sin x}{x} + y\ln x\cos x$ | A1 | o.e.; accept in terms of $x$ only; ISW after correct answer |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $\frac{dy}{dx}=0$, divides by $y$ or $x^{\sin x}$ | M1 | $\frac{dy}{dx}$ must be in correct form |
| $\frac{\sin x}{\cos x} + x\ln x = 0 \Rightarrow \tan x + x\ln x = 0$ | A1* | Must show intermediate line $\frac{\sin x}{\cos x}+x\ln x=0$; can be implied by $\sin x + x\ln x\cos x = 0 \div\cos x$ |
---6. A curve $C$ has equation
$$y = x ^ { \sin x } \quad x > 0 \quad y > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find, by firstly taking natural logarithms, an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
\item Hence show that the $x$ coordinates of the stationary points of $C$ are solutions of the equation
$$\tan x + x \ln x = 0$$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2020 Q6 [7]}}