Q.Differentiate w.r.t. $x$: $\cos\left(\tan\sqrt{x + 1}\right)$.

Concept: Chain Rule – we differentiate from the outermost function inward.

Let $y = \cos(u)$, where $u = \tan(v)$ and $v = \sqrt{x+1}$.

Step 1: Differentiate the outermost:

$\frac{dy}{du} = -\sin(u) = -\sin\left(\tan\sqrt{x+1}\right)$.

Step 2: Differentiate the middle:

$\frac{du}{dv} = \sec^2(v) = \sec^2\left(\sqrt{x+1}\right)$.

Step 3: Differentiate the innermost:

$\frac{dv}{dx} = \frac{1}{2\sqrt{x+1}}$.

Multiply all three derivatives: …

The problem asks us to differentiate a composition of three functions: cosine, tangent, and square root. When you have a function inside a function inside another function, the Chain Rule is your only tool — and it works exactly the same way as for two functions, just applied repeatedly.

Think of it like peeling an onion: start from the outermost layer and work inward. Each layer you peel gives you a derivative factor, and you multiply them all together.

Let’s name the layers clearly:

• Outermost: $\cos(\text{something})$
• Middle: $\tan(\text{something})$
• Innermost: $\sqrt{x+1}$

We’ll differentiate step by step.

1. Differentiate the outermost function. The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$. Here $u = \tan\sqrt{x+1}$. So the first factor is:

2. Multiply by the derivative of the middle function. Now we need the derivative of $\tan(v)$ with respect to $v$, where $v = \sqrt{x+1}$. The derivative of $\tan v$ is $\sec^2 v$. So the second factor is:

3. Multiply by the derivative of the innermost function. Finally, differentiate $\sqrt{x+1}$ with respect to $x$. Write it as $(x+1)^{1/2}$. Its derivative is:

4. Multiply all three factors together. The complete derivative is: …

Method: Chain Rule for Three Nested Non-Standard Functions

Use this method for a composition of three unlike functions stacked on each other — such as cosine of tangent of a square root — where each layer uses a different derivative rule.

Steps

Step 1: Name each layer from outside in

For $\cos\big(\tan\sqrt{x+1}\big)$: outermost is $\cos(\cdot)$, middle is $\tan(\cdot)$, innermost is $\sqrt{x+1}$. Writing $y=\cos u$, $u=\tan v$, $v=\sqrt{x+1}$ makes the structure explicit.

Step 2: Differentiate the outermost layer

Remember the negative sign — this is where it is easiest to lose.

Step 3: Differentiate the middle layer

Step 4: Differentiate the innermost layer …

Common Mistakes

Mistake 1: Multiplying only two of the three chain-rule factors

Why it's wrong: with three genuinely different layers (cosine, tangent, square root), it is easy to differentiate the outer two correctly and forget the innermost $\dfrac{1}{2\sqrt{x+1}}$ factor entirely, or vice versa. Correct approach: explicitly count the layers before starting, and check at the end that the final answer contains exactly one derivative factor per layer.

Mistake 2: Dropping the negative sign from differentiating cosine …