# Derivative A2

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https://algebrica.org/exercises/derivative-a-2/
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This exercise requires calculating the derivative of a [composite power function](../derivative-of-composite-power-functions.md) of the form \\( f(x)^{g(x)} \\).

Let’s consider the function \\( y = x^{ln(x)} \\), and calculate its derivative.

* * *

First, let’s rewrite the function by applying the logarithm to both sides:

\\[
\\ln y = \\ln(x)^{ln(x)}
\\]

For the properties of logarithms \\(\\log\_a(b^c) = c \\cdot \\log\_a(b)\\)

The equality can be rewritten as:

\\[
\\ln y = \\ln(x) \\cdot \\ln(x)
\\]

* * *

Since \\(\\ln y\\) is a composite function, its derivative is

\\[
\\frac{1}{y} \\cdot y’
\\]

Let’s compute the derivative for the element on the right-hand side of the equality \\(ln(x) \\cdot \\log(x)\\):

\\[
\\frac{1}{x} \\cdot ln(x) + \\frac{1}{x} \\cdot ln(x)
\\]

We obtain:

\\[
\\frac{1}{y} \\cdot y’ = \\frac{1}{x} \\cdot ln(x) + \\frac{1}{x} \\cdot ln(x)
\\]

* * *

The equality can be rewritten as:

\\[
y’ = y \\cdot \\frac{2}{x} \\cdot ln(x)
\\]

Since \\(y = x^{ln(x)}\\), we have:

\\[
y’ = x^{ln(x)} \\cdot \\frac{2}{x} \\cdot ln(x)
\\]

Thus, the derivative of y = x^{ln(x)} is equal to:

\\[
y’ = 2x^{ln(x)} \\cdot \\frac{ln(x)}{x}
\\]