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

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

Let $y = \log(\log x)$. The outer function is $\log u$ with $u = \log x$.

Step 1: Differentiate the outer function:

$\frac{dy}{du} = \frac{1}{u} = \frac{1}{\log x}$.

Step 2: Differentiate the inner function:

$\frac{du}{dx} = \frac{1}{x}$. …

The key idea here is the chain rule. Whenever you have a function nested inside another function — like $\log(\log x)$ — you differentiate from the outside in. Think of it as peeling an onion: the outermost layer is the logarithm of something, then inside that is another logarithm of $x$.

The chain rule says: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Here, $f$ is the outer log and $g$ is the inner log.

Let’s work through it step by step.

1. Identify the outer and inner functions.

   We have $y = \log(\log x)$.

   • Outer function: $\log(u)$, where $u = \log x$.
   • Inner function: $u = \log x$.

2. Differentiate the outer function with respect to its argument.

   The derivative of $\log(u)$ with respect to $u$ is $\frac{1}{u}$.

   So, $\frac{dy}{du} = \frac{1}{u} = \frac{1}{\log x}$.

3. Differentiate the inner function with respect to $x$.

   The derivative of $\log x$ with respect to $x$ is $\frac{1}{x}$.

   So, $\frac{du}{dx} = \frac{1}{x}$.

4. Apply the chain rule.

   Multiply the two derivatives: …

Method: The Chain Rule for a Composite Function

Whenever a function is "wrapped inside" another function — $f(g(x))$ — differentiate the outer function first (with respect to its own argument), then multiply by the derivative of the inner function.

Steps

Step 1: Identify the outer function and the inner function

Write $y=f(u)$ where $u=g(x)$ is everything "inside" the outermost operation.

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

Use the standard derivative rule for whatever the outer function is (log, power, trig, exponential, ...), keeping $u$ untouched.

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

Step 4: Multiply the two results

$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.$ …

Common Mistakes

Mistake 1: Stopping after differentiating only the outer $\log$, writing $\frac{1}{\log x}$ as the final answer.

Why it's wrong: this omits the second chain-rule factor $\frac{1}{x}$ that comes from differentiating the inner $\log x$ — any function nested inside another needs both layers differentiated. Correct approach: explicitly identify and differentiate the inner function as its own step before multiplying.

Mistake 2: Confusing $\log(\log x)$ with $(\log x)^2$ or another algebraic combination. …