Common Mistakes in differentiation and trigonometry (1 Viewer)

StarKiller2196

New Member
What are some common mistakes people make on questions such as:
differentiation
Auxiliary Angle questions
Derivative Graphing
Thanks

Pedro123

Member
Differentiation - Usually nothing, but if you don't have the identities down pat, then it can be annoying (i.e. sec(x), cot(x), inverse functions, etc)
Auxiliary Angle - Not remembering all the different formulas (Many people only remember a few since some can be derived, however in an exam it is much better if you remember everything). Also t substitution can be annoying
Derivative graphing - this is mostly just stupid misinterpretation issues, i.e. not realising when a curve has a reducing derivative that is still positive, points of inflextion

fan96

617 pages
This is one of the more common differentiation mistakes.

$\bg_white \frac{d}{dx} x^x \overset{?}=x \cdot x^{x-1}$

For trigonometry in general, people make a lot of mistakes.

$\bg_white \cot x \overset{?}= \frac{1}{\tan x}$

$\bg_white \sin^{-1}( \sin x ) \overset{?}= x$, or equivalently, $\bg_white \sin x = y \overset{?} \implies x = \sin^{-1} y$

A lot of these involve division by zero.

\bg_white \begin{aligned} (\sin x )(\cos x - 1) &= \sin^2x \\ \implies \cos x - 1 & \overset{?}= \sin x \end{aligned}

Many such examples are nuanced.
One example discussed here recently was the CSSA 2019 trial paper, where one of the questions was

$\bg_white \text{Show that }\frac{2 \tan x}{1 + \tan^2 x} = \sin 2x.$

This equality is not even true to begin with, for reasons discussed in that thread.

The take-away here is that you should never divide by anything unless you know that it will never be zero.

CM_Tutor

Well-Known Member
Mistakes can be made with auxilliary angle method questions by not adjusting the domain (or doing so incorrectly), and thereby losing solutions