Challenging +1.2 This requires applying the first principles definition to a trigonometric function, using the compound angle formula and standard limits (sin h/h → 1, (cos h - 1)/h → 0). While the method is bookwork that students learn, it's a multi-step proof requiring careful algebraic manipulation and knowledge of trigonometric limits, making it moderately harder than average routine differentiation questions.
\(\cos(x+\delta x)-\cos x = \cos x\cos\delta x - \sin x\sin\delta x - \cos x\)
B1
Allow \(h\) or other letter for \(\delta x\) throughout
\(\lim_{\delta x\to 0}\dfrac{\cos x\cos\delta x - \sin x\sin\delta x - \cos x}{\delta x}\)
M1
or \(\lim_{\delta x\to 0}\dfrac{\cos(x+\delta x)-\cos x}{\delta x}\) or may be seen later. Must include \(\lim_{\delta x\to 0}\)
as \(\delta x\to 0\): \(\cos\delta x\to 1\) or \(1-\frac{(\delta x)^2}{2}\)
Allow \(\cos\delta x=1\) for small \(\delta x\) (or \(1-\frac{(\delta x)^2}{2}\))
and \(\dfrac{\sin\delta x}{\delta x}\to 1\) or \(\sin\delta x\to\delta x\)
M1
Allow \(\sin\delta x=\delta x\) for small \(\delta x\). Both must be explicitly stated for M1. If not stated but implied, M0, but can still possibly gain final A1
\(\left(\lim_{\delta x\to 0}\dfrac{\cos x - \sin x\,\delta x - \cos x}{\delta x}\right) = -\sin x\)
A1
Dep on at least B1M1 gained, and approximations either seen explicitly or seen substituted, and nothing incorrect seen. NB. \(\cos x - \sin x - \cos x = -\sin x\) is incorrect and scores A0
## Question 7:
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cos(x+\delta x)-\cos x = \cos x\cos\delta x - \sin x\sin\delta x - \cos x$ | **B1** | Allow $h$ or other letter for $\delta x$ throughout |
| $\lim_{\delta x\to 0}\dfrac{\cos x\cos\delta x - \sin x\sin\delta x - \cos x}{\delta x}$ | **M1** | or $\lim_{\delta x\to 0}\dfrac{\cos(x+\delta x)-\cos x}{\delta x}$ or may be seen later. Must include $\lim_{\delta x\to 0}$ |
| as $\delta x\to 0$: $\cos\delta x\to 1$ or $1-\frac{(\delta x)^2}{2}$ | | Allow $\cos\delta x=1$ for small $\delta x$ (or $1-\frac{(\delta x)^2}{2}$) |
| and $\dfrac{\sin\delta x}{\delta x}\to 1$ or $\sin\delta x\to\delta x$ | **M1** | Allow $\sin\delta x=\delta x$ for small $\delta x$. Both must be explicitly stated for M1. If not stated but implied, M0, but can still possibly gain final A1 |
| $\left(\lim_{\delta x\to 0}\dfrac{\cos x - \sin x\,\delta x - \cos x}{\delta x}\right) = -\sin x$ | **A1** | Dep on at least B1M1 gained, and approximations either seen explicitly or seen substituted, and nothing incorrect seen. NB. $\cos x - \sin x - \cos x = -\sin x$ is incorrect and scores A0 |
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