Derivative A2

This exercise requires calculating the derivative of a composite power function 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:

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