Question

If we have an expression as ${{x}^{m{{x}^{m{{x}^{m{{x}^{mx\ldots }}}}}}}}={{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$, then $\dfrac{dy}{dx}$ is equal to:
1. $\dfrac{y}{x}$
2. $\dfrac{x}{y}$
3. $\dfrac{my}{mx}$
4. $\dfrac{ny}{mx}$

Answer 1

Hint: For solving this question you should know about the differentiation and differentiation of a function to infinite power. In this problem first we will put the function as it is given and then make it equal to one by dividing sufficient terms on both sides and then shorten this as much as possible. And then at last we will differentiate that function to get the answer.

Complete step-by-step solution:
According to the question the function is given to us as ${{x}^{m{{x}^{m{{x}^{m{{x}^{mx\ldots }}}}}}}}={{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$and asked to find $\dfrac{dy}{dx}$ for this. So, as we can see here that the function is given as,
${{x}^{m\left( {{x}^{m{{x}^{m{{x}^{mx\ldots }}}}}} \right)}}={{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$
As we can see that the power is going till infinite, so we can reduce ${{x}^{m{{x}^{m{{x}^{m{{x}^{mx\ldots }}}}}}}}={{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$ to our expression and we can write it as,
${{x}^{m\left( {{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}} \right)}}={{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$
Now for making it’s value 1, we divide both the sides by ${{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}$.
So, we get
$\begin{align}
  & \dfrac{{{x}^{m{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}}{{{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}}=\dfrac{{{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}}{{{y}^{n{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}} \\
 & \Rightarrow \dfrac{{{\left( {{x}^{m}} \right)}^{{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}}{{{\left( {{y}^{n}} \right)}^{{{y}^{n{{y}^{n{{y}^{ny\ldots }}}}}}}}}=1 \\
\end{align}$
Now it can be further written as,
${{\left( \dfrac{{{x}^{m}}}{{{y}^{n}}} \right)}^{{{y}^{n{{y}^{ny\ldots }}}}}}=1$
And here we can easily remove this power, because we know that 1 to the power anything is always 1. So, we get
$\dfrac{{{x}^{m}}}{{{y}^{n}}}=1$
We now multiply both the sides by ${{y}^{n}}$, so we will get,
${{x}^{m}}={{y}^{n}}$
And now differentiate the above equation with respect to x, so we will get as follows,
$\begin{align}
  & m{{x}^{m-1}}=n{{y}^{n-1}}.\dfrac{dy}{dx} \\
 & \Rightarrow \dfrac{dy}{dx}=\left( \dfrac{m}{n} \right)\dfrac{{{x}^{m-1}}}{{{y}^{n-1}}} \\
\end{align}$
So, this is the final value of $\dfrac{dy}{dx}$.

Note: For solving these types of questions you have to know if the power of any variable is going till infinite, then always reduce that by making any mathematical operation of division or multiplication.