Derivative A2

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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)

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

y’ = 2x^{ln(x)} \cdot \frac{ln(x)}{x}