Q.If $f(x) = |\cos x - \sin x|$, then $f'\left(\dfrac{\pi}{3}\right) =$ __________.

Concept: Derivative Evaluation — The key is to first determine the sign of the expression inside the absolute value at the given point, then differentiate accordingly.

Step 1: Evaluate $\cos x - \sin x$ at $x = \frac{\pi}{3}$.

Step 2: Since the expression is negative, $f(x) = -(\cos x - \sin x) = \sin x - \cos x$ near $x = \frac{\pi}{3}$. …

Fix the branch. At $x=\dfrac{\pi}{3}$, $\cos\dfrac{\pi}{3}=\dfrac12$ and $\sin\dfrac{\pi}{3}=\dfrac{\sqrt3}{2}$, so

Near this point $|\cos x-\sin x|=-(\cos x-\sin x)=\sin x-\cos x$.

Differentiate that branch.

$f'(x)=\cos x+\sin x.$ …

Method: Differentiating an Absolute Value Function at a Specific Point (Sign-Check Method)

This method solves "find $f'(a)$ where $f(x)=|g(x)|$" problems by determining the sign of the expression inside the modulus at the given point, then differentiating the corresponding branch.

Steps

Step 1: Evaluate the inside expression at the given point

Compute $g(a)$ — here $g(x)=\cos x-\sin x$ — at the specific point where the derivative is asked for.

Step 2: Determine its sign

If $g(a)>0$, continuity of $g$ guarantees it stays positive near $a$, so $|g(x)|=g(x)$ locally. If $g(a)<0$, it stays negative near $a$, so $|g(x)|=-g(x)$ (the negative of $g$) locally.

Step 3: Differentiate the branch that applies

$f'(x)=\begin{cases} g'(x), & g(a)>0,\\ -g'(x), & g(a)<0.\end{cases}$ …

Common Mistakes

Mistake 1: Assuming the inside expression is positive by habit

Having just solved a similar problem where $\cos x>0$ at the given point, a student may skip the sign check here and differentiate $\cos x-\sin x$ directly as $-\sin x-\cos x$. Why it's wrong: at $x=\frac{\pi}{3}$, $\cos\frac{\pi}{3}-\sin\frac{\pi}{3}=\frac{1-\sqrt3}{2}<0$ — the expression is actually NEGATIVE here, so the absolute value flips its sign, not preserves it. Skipping the check gives $-\frac{1+\sqrt3}{2}$, the wrong sign on the final answer. Correct approach: always evaluate the inside expression numerically at the given point before deciding which branch of $|u|$ applies.

Mistake 2: Arithmetic slip comparing $\cos\frac{\pi}{3}$ and $\sin\frac{\pi}{3}$ …