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 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,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.