Q.If $y = (\cos x)^{(\cos x)^{(\cos x)^{\cdots\infty}}}$, show that $\dfrac{dy}{dx} = \dfrac{y^2 \tan x}{y \log \cos x - 1}$.

Concept: Implicit Differentiation — the infinite tower means $y = (\cos x)^y$.

Step 1: Write the given infinite power tower as

$y = (\cos x)^y$.

Step 2: Take natural logarithms on both sides:

$\log y = y \log(\cos x)$.

Step 3: Differentiate implicitly with respect to $x$:

$\frac{1}{y} \frac{dy}{dx} = \frac{dy}{dx} \log(\cos x) + y \cdot \frac{-\sin x}{\cos x}$.

Step 4: Simplify using $\tan x = \frac{\sin x}{\cos x}$ and collect $\frac{dy}{dx}$ terms:

$\frac{1}{y} \frac{dy}{dx} - \log(\cos x) \frac{dy}{dx} = -y \tan x$

$\Rightarrow \frac{dy}{dx} \left( \frac{1}{y} - \log(\cos x) \right) = -y \tan x$ …

The key insight: when you see an infinite tower like $a^{a^{a^{\cdots}}}$, it means the exponent is the same expression all over again. So if $y$ equals the whole tower, then $y$ also equals $(\cos x)^y$. This self-referential equation is the heart of the solution — it lets us avoid dealing with the infinite chain directly.

Let’s walk through it.

1. Set up the self-referential equation Since the tower goes on forever, the exponent of the first $\cos x$ is itself the entire tower. Therefore:

This is valid only where the tower converges (typically for $e^{-e} \le \cos x \le e^{1/e}$, but we assume the domain is such that the expression is well-defined).

2. Take the natural logarithm of both sides This brings the exponent down:

Notice: $\log(\cos x)$ is defined when $\cos x > 0$, which is a natural domain restriction.

3. Differentiate implicitly with respect to $x$ Both sides are functions of $x$, and $y$ is a function of $x$. Differentiate:
   • Left side: $\frac{d}{dx} \log y = \frac{1}{y} \cdot \frac{dy}{dx}$
   • Right side: use the product rule on $y \cdot \log(\cos x)$:

 The derivative of $\log(\cos x)$ is $\frac{-\sin x}{\cos x} = -\tan x$.

So we have:

4. Collect terms with $\frac{dy}{dx}$ Bring the $\frac{dy}{dx} \log(\cos x)$ term to the left:

Factor out $\frac{dy}{dx}$:

5. Solve for $\frac{dy}{dx}$ Multiply both sides by $y$ to clear the fraction inside the bracket: …

Method: Solving Infinite Power Towers by Self-Reference

Use this method for any expression of the form $y=a(x)^{a(x)^{a(x)^{\cdots}}}$ (an infinite exponent tower). The trick is recognising that the tower, being infinite, repeats itself inside its own exponent.

Steps

Step 1: Replace the infinite tower with a self-referential equation

Since the tower never ends, the exponent on the very first factor is itself the entire tower — which is just $y$ again:

For this problem, $a(x)=\cos x$, so $y=(\cos x)^y$. This single algebraic step replaces an "infinite" object with an ordinary implicit equation.

Step 2: Take the natural logarithm of both sides

This is necessary because $y$ sits in the exponent — logarithmic differentiation is the standard tool whenever the differentiation variable appears as an exponent.

Step 3: Differentiate implicitly with respect to $x$

The left side needs the chain rule; the right side is a product, so needs the product rule, and $\log(\cos x)$ itself needs the chain rule: …

Common Mistakes

Mistake 1: Trying to differentiate the infinite tower "layer by layer"

Some students see the infinite exponent tower and attempt to peel off one layer of $\cos x$ at a time, which never terminates. Why it's wrong: an infinite tower can't be differentiated term-by-term — it must first be collapsed using its self-referential property. Correct approach: recognize that because the tower repeats forever, the whole tower equals $y=(\cos x)^y$, replacing the infinite expression with one clean implicit equation.

Mistake 2: Ignoring the domain/convergence condition

Students often don't pause to note that $\log(\cos x)$ requires $\cos x>0$, and that the infinite tower only converges for a restricted range of values. Why it's wrong: proceeding without this in mind can mask where the final formula is actually valid. Correct approach: note that the derivation assumes $\cos x>0$ so $\log(\cos x)$ is defined.

Mistake 3: Sign error differentiating $\log(\cos x)$, or dropping the product rule on $y\log(\cos x)$ …