Q.Find $\frac{dy}{dx}$ in the following: $\cos(\sqrt{x})$

Concept: Chain Rule — differentiate the outer function, then multiply by the derivative of the inner function.

Let $y = \cos(\sqrt{x})$.

The outer function is $\cos(u)$, where $u = \sqrt{x} = x^{1/2}$.

Step 1: $\frac{dy}{du} = -\sin(u) = -\sin(\sqrt{x})$.

Step 2: $\frac{du}{dx} = \frac{1}{2\sqrt{x}}$. …

The core idea here is the chain rule — the essential tool when one function is "inside" another. Think of it like peeling an onion: you differentiate the outer layer first, leaving the inner part untouched, then multiply by the derivative of the inner layer. For $\cos(\sqrt{x})$, the outer function is $\cos(u)$ and the inner function is $u = \sqrt{x}$. The chain rule says:

Now let's work through it step by step.

1. Identify the composition.

   We have $y = \cos(\sqrt{x})$. Let $u = \sqrt{x}$. Then $y = \cos(u)$. This makes the chain rule straightforward: we need $\frac{dy}{du}$ and $\frac{du}{dx}$.

2. Differentiate the outer function.

   The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$. So:

3. Differentiate the inner function. The inner function is $u = \sqrt{x} = x^{1/2}$. Its derivative is:

4. Apply the chain rule. Multiply the two derivatives:

5. Substitute back $u = \sqrt{x}$. Replace $u$ with $\sqrt{x}$ to express everything in terms of $x$: …

Method: The Two-Layer Chain Rule

Use this method for the most basic chain-rule case: a function with exactly one "outer" function wrapped around one "inner" function, $y = f(g(x))$.

Steps

Step 1: Name the outer and inner function explicitly

Write $y = f(u)$ where $u = g(x)$. Resist the temptation to differentiate "in your head" — writing $u$ down explicitly prevents accidentally skipping a layer.

Step 2: Differentiate the outer function with respect to $u$

Use the standard derivative rule for whatever $f$ is (trig, exponential, power, etc.), keeping $u$ as a single symbol rather than substituting its formula in yet.

Step 3: Differentiate the inner function with respect to $x$ …

Common Mistakes

Mistake 1: Forgetting to multiply by the derivative of the inner function $\sqrt{x}$

Why it's wrong: writing $\frac{d}{dx}\cos(\sqrt x) = -\sin(\sqrt x)$ and stopping there only accounts for the outer cosine layer — the chain rule always requires an extra factor from the inner function, here $\frac{1}{2\sqrt x}$. Correct approach: explicitly identify $u=\sqrt x$ before differentiating, so the multiplication by $\frac{du}{dx}$ isn't skipped.

Mistake 2: Writing $\frac{d}{dx}\sqrt x = \frac{1}{\sqrt x}$ instead of $\frac{1}{2\sqrt x}$ …