Q.Differentiate w.r.t. $x$: $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right),\ -\dfrac{\pi}{4} < x < \dfrac{\pi}{4}$.

Concept: Domain Of Composite Function — the given interval ensures the inner expression lies in $[-1,1]$, so the inverse cosine is defined.

Step 1: Simplify the inner function.

Step 2: For $-\frac{\pi}{4} < x < \frac{\pi}{4}$, we have $0 < x + \frac{\pi}{4} < \frac{\pi}{2}$, so $\sin(x + \pi/4)$ is positive and in $(0,1)$. Hence the function becomes

$y = \cos^{-1}\left(\sin\left(x + \frac{\pi}{4}\right)\right)$ …

We are asked to differentiate $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right)$ with respect to $x$, for $-\dfrac{\pi}{4} < x < \dfrac{\pi}{4}$.

The expression inside the inverse cosine looks like it could be a single trigonometric function. Recall the identity: $\sin A \cos B + \cos A \sin B = \sin(A+B)$. If we factor $\frac{1}{\sqrt{2}}$, we can write $\frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}}$. Notice that $\frac{1}{\sqrt{2}} = \sin \frac{\pi}{4} = \cos \frac{\pi}{4}$. So:

This is exactly $\sin\left(x + \frac{\pi}{4}\right)$.

So the function becomes:

Now we need to differentiate this. But $\cos^{-1}$ and $\sin$ are related: $\cos^{-1}(\sin \theta) = \frac{\pi}{2} - \theta$, provided $\theta$ lies in the range where this holds. Let's check the domain. …

Method: Auxiliary-Angle Identity to Collapse $a\sin x + b\cos x$ Before Differentiating

Whenever the argument of an inverse trig function is a combination like $a\sin x + b\cos x$ (or that combination divided by a constant), rewrite it as a single sine or cosine first — direct quotient-rule/chain-rule differentiation of the raw expression is far messier and more error-prone.

Steps

Step 1: Recognise the $a\sin x + b\cos x$ pattern

Here the argument is $\dfrac{\sin x + \cos x}{\sqrt2}$, i.e. $a=b=1$ divided by $\sqrt2=\sqrt{a^2+b^2}$.

Step 2: Rewrite as a single sine using the addition formula

Since $\dfrac{1}{\sqrt2} = \sin\dfrac\pi4 = \cos\dfrac\pi4$,

Step 3: Use the complementary-angle identity, checking the range carefully

$\cos^{-1}(\sin\theta) = \frac\pi2-\theta \qquad\text{valid for } \theta\in\left[-\frac\pi2,\frac\pi2\right]$ …

Common Mistakes

Mistake 1: Applying $\cos^{-1}(\sin\theta)=\frac\pi2-\theta$ without checking the range of $\theta$

Why it's wrong: this identity only holds when $\theta\in[-\pi/2,\pi/2]$; outside that range the correct simplification involves an extra $\pi$ shift, and blindly applying the formula gives a wrong function before you even differentiate. Correct approach: always verify the given domain maps $\theta=x+\pi/4$ into the valid interval before using the identity.

Mistake 2: Trying to differentiate the original expression directly with the chain and quotient rules …