Animated Solution for Mathematics - Differentiation: If $x=e^{y+e^{y+e^{y+\dots \infty }}},x>0$, then $\frac{dy}{dx}$ is

The expression provided is $x=e^{y+e^{y+e^{y+\dots \infty }}}$. At first glance, the infinite exponent feels like a trap, but in mathematics, the void is often just a mirror.

Imagine you are standing at the edge of an infinite staircase. If you take one step down, you are still standing on an infinite staircase. This is the concept of self-similarity.

Look closely at the exponent of the first $e$. The term $e^{y+e^{y+\dots \infty }}$ is exactly the same as our original expression for $x$. Because the series goes to infinity, removing the first layer changes nothing.

We can replace that entire infinite repeating part simply with $x$. Suddenly, our scary infinite equation collapses into a very neat and simple form:

We have effectively eliminated the infinity.

Now, we have $x=e^{y+x}$. Our goal is to find $\frac{dy}{dx}$, but $y$ is trapped in the exponent. To bring it down to the ground, we use the natural logarithm, $\ln$.

By taking $\ln$ on both sides, we get:

Using the fundamental property of logarithms, $\ln (e^a)=a$, the right side simplifies beautifully. The $e$ and the $\ln$ cancel out, leaving us with:

This is the bridge that connects the exponential world to the linear world.

Now, we are ready for the final act. We need to differentiate both sides with respect to $x$ by applying the derivative operator $\frac{d}{dx}$ to both sides:

On the left, the derivative of $\ln (x)$ is a standard result: $\frac{1}{x}$. On the right, we differentiate term by term. The derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$, and the derivative of $x$ with respect to $x$ is $1$.

This yields the equation:

We are almost there. We need to isolate $\frac{dy}{dx}$ by shifting the $1$ to the left side:

To get the final answer, we take the common denominator $x$, which gives us:

And there it is! A problem that seemed to stretch to infinity has been solved in a few simple steps. The key takeaway for your JEE preparation is this: whenever you see an infinite recursive function, look for the repeating block, substitute it with the original variable, and watch the complexity vanish.