Q.Differentiate w.r.t. $x$: $\log\left[\log(\log x^5)\right]$.

Concept: Chain Rule – differentiate the outermost function, then multiply by the derivative of the inner function, repeating until the innermost is reached.

Step 1: Let $y = \log\left[\log(\log x^5)\right]$. The outermost function is $\log(\cdot)$, so

$\frac{dy}{dx} = \frac{1}{\log(\log x^5)} \cdot \frac{d}{dx}\left[\log(\log x^5)\right]$.

Step 2: Differentiate $\log(\log x^5)$:

$\frac{d}{dx}\left[\log(\log x^5)\right] = \frac{1}{\log x^5} \cdot \frac{d}{dx}(\log x^5)$. …

The key idea here is the chain rule — when you have a function nested inside another function, you differentiate from the outermost layer inward, multiplying the derivatives at each step. This problem has three layers of nesting: the outer $\log$, then another $\log$, then $\log(x^5)$, and finally $x^5$ itself. Each layer is a function of the one below it.

Think of it like peeling an onion: start with the outermost peel (the first $\log$), differentiate it, then move to the next layer, and so on, until you reach the innermost $x^5$. At each step, you leave the inner part untouched while differentiating the outer part.

Let’s work through it step by step.

1. Identify the structure.

   Let $y = \log\left[\log(\log x^5)\right]$. Here, $\log$ means the natural logarithm (base $e$), as is standard in calculus. The innermost part is $x^5$, then $\log(x^5)$, then $\log(\log x^5)$, and finally the outermost $\log$.

2. Apply the chain rule — first layer.

   The derivative of $\log(u)$ with respect to $u$ is $\frac{1}{u}$. So, treating $u = \log(\log x^5)$, we have:

3. Second layer — differentiate $\log(\log x^5)$. Now let $v = \log x^5$. Then $\frac{d}{dx}[\log v] = \frac{1}{v} \cdot \frac{dv}{dx}$. So:

4. Third layer — differentiate $\log(x^5)$. Here, $\log(x^5) = 5 \log x$ (using the logarithm power rule). Its derivative is:

Alternatively, you can use the chain rule directly: derivative of $\log(x^5)$ is $\frac{1}{x^5} \cdot 5x^4 = \frac{5}{x}$. Same result.

5. Combine all layers. Multiply everything together: …

Method: Chain Rule for Multiply-Nested Logarithms — Peel from Outside In

Use this method whenever a function is built from several logarithms nested inside one another, such as $\log[\log(\log x^5)]$.

Steps

Step 1: List the layers from outermost to innermost

Identify each nested function separately before differentiating anything. Here: outermost is $\log(\cdot)$, next is $\log(\cdot)$ again (applied to $\log x^5$), and innermost is $\log(x^5)$, which itself depends on $x^5$.

Step 2: Differentiate the outermost layer, leaving the inside untouched

This produces the factor $\dfrac{1}{\log(\log x^5)}$, multiplied by the derivative of the next layer in.

Step 3: Repeat for each successive layer

Differentiate $\log(\log x^5)$ the same way, treating $v=\log x^5$ as its inner function, producing another factor $\dfrac{1}{\log x^5}$ multiplied by $\dfrac{d}{dx}(\log x^5)$.

Step 4: Differentiate the innermost layer using the logarithm power rule …

Common Mistakes

Mistake 1: Stopping after differentiating only one or two of the nested logs

Why it's wrong: with three layers of nesting, skipping a layer (for example differentiating the outer log and jumping straight to $\frac{5}{x}$ without the middle $\frac{1}{\log x^5}$ factor) drops an entire reciprocal factor from the answer. Correct approach: explicitly list every layer before starting, and confirm the final answer has exactly one factor per layer.

Mistake 2: Confusing $\log(x^5)$ with $(\log x)^5$ or $\log^2(x^5)$

Why it's wrong: the brackets in $\log[\log(\log x^5)]$ specify that each $\log$ applies to the entire result of the previous step, not that anything is being raised to a power; misreading the nesting leads to differentiating a completely different function. Correct approach: carefully identify what each logarithm's argument actually is, from the innermost bracket outward, before writing any derivative. …