Standard +0.3 This is a straightforward quotient rule application with trigonometric functions, requiring algebraic manipulation to match the given form. While it involves multiple steps (differentiation, simplification, and finding integer constants), the techniques are standard for P3 level with no novel insight required. Slightly easier than average due to the 'show that' format providing the target expression.
6.
$$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$
Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$
where \(a , b\) and \(c\) are integers to be found.
\(\frac{d}{dx}(\cos x + \sin x) = -\sin x + \cos x\)
B1
Correct differentiation; can be scored in workings for quotient rule or even in denominator of incorrect attempt
\(\frac{dy}{dx} = \frac{(\cos x + \sin x)(3\cos x) - (-\sin x + \cos x)(2 + 3\sin x)}{(\cos x + \sin x)^2}\)
M1
Differentiates using quotient or product rule; for quotient rule look for \(\frac{(\cos x+\sin x)(\pm a\cos x)-(\pm\sin x \pm \cos x)(2+3\sin x)}{(\cos x+\sin x)^2}\)
Fully correct derivative
A1
\(= \frac{3\cos^2 x + 3\sin x\cos x + 2\sin x + 3\sin^2 x - 2\cos x - 3\sin x\cos x}{\cos^2 x + 2\sin x\cos x + \sin^2 x} = \frac{3 + 2\sin x - 2\cos x}{1 + 2\sin x\cos x}\)
M1
Expands numerator or denominator and applies \(\sin^2 x + \cos^2 x = 1\) at least once
\(= \frac{3 + 2\sin x - 2\cos x}{1 + 2\sin x\cos x} \times \frac{\sec x}{\sec x} = \frac{3\sec x + 2\sec x\sin x - 2}{\sec x + 2\sin x}\)
M1
Multiplies through by \(\sec x\) in numerator and denominator; not dependent, may be scored before previous M
\(= \frac{2\tan x + 3\sec x - 2}{\sec x + 2\sin x}\)
A1
Correct answer; terms may be in different order; allow minor notation slips
# Question 6:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{d}{dx}(\cos x + \sin x) = -\sin x + \cos x$ | B1 | Correct differentiation; can be scored in workings for quotient rule or even in denominator of incorrect attempt |
| $\frac{dy}{dx} = \frac{(\cos x + \sin x)(3\cos x) - (-\sin x + \cos x)(2 + 3\sin x)}{(\cos x + \sin x)^2}$ | M1 | Differentiates using quotient or product rule; for quotient rule look for $\frac{(\cos x+\sin x)(\pm a\cos x)-(\pm\sin x \pm \cos x)(2+3\sin x)}{(\cos x+\sin x)^2}$ |
| Fully correct derivative | A1 | |
| $= \frac{3\cos^2 x + 3\sin x\cos x + 2\sin x + 3\sin^2 x - 2\cos x - 3\sin x\cos x}{\cos^2 x + 2\sin x\cos x + \sin^2 x} = \frac{3 + 2\sin x - 2\cos x}{1 + 2\sin x\cos x}$ | M1 | Expands numerator or denominator and applies $\sin^2 x + \cos^2 x = 1$ at least once |
| $= \frac{3 + 2\sin x - 2\cos x}{1 + 2\sin x\cos x} \times \frac{\sec x}{\sec x} = \frac{3\sec x + 2\sec x\sin x - 2}{\sec x + 2\sin x}$ | M1 | Multiplies through by $\sec x$ in numerator and denominator; not dependent, may be scored before previous M |
| $= \frac{2\tan x + 3\sec x - 2}{\sec x + 2\sin x}$ | A1 | Correct answer; terms may be in different order; allow minor notation slips |
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6.
$$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$
Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$
where $a , b$ and $c$ are integers to be found.\\
\hfill \mbox{\textit{Edexcel P3 2022 Q6 [6]}}