Math CentralQuandaries & Queries


Question from Nafis:

differentiate y = x^x^x

Hi Nafis,

First you need to decide if this is

\[y = x^{\left(x^x\right)} \mbox{ or } y = \left(x^x\right)^{x}.\]

If you don't see that these are different calculate

\[\left(2^3\right)^3 \mbox{ and } 2^{\left(3^3\right)}.\]

In either case you have expressions of the form

\[h(x) = f(x)^{g(x)}.\]

To differentiate $h(x)$ first take the Natural Logarithm of both sides to get

\[\ln(h(x)) = \ln\left(f(x)^{g(x)}\right) = g(x) \ln(f(x)).\]

now differentiate with respect to $x$ to get

\[\frac{h^{\prime}(x)}{h(x)} = g^{\prime}(x) \ln(f(x)) + g(x) \frac{f^{\prime}(x)}{f(x)}.\]

Solve for $h^{\prime}(x).$

This expression should allow you to solve your problem,

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