Derivative A1

This exercise requires calculating the derivative of a composite power function of the form \( f(x)^{g(x)} \).

Let’s consider the function \( y = x^{2cosx} \), and calculate its derivative.

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

\[\ln y = \ln(x^{2cosx})\]

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

The equality can be rewritten as:

\[\ln y = 2cosx \cdot \ln(x)\]

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 \(2cosx \cdot \log(x)\):

\[-2\sin(x) \cdot \ln(x) + \frac{2\cos(x)}{x} \]

We obtain:

\[ \frac{1}{y} \cdot y’ = -2\sin(x) \cdot \ln(x) + \frac{2\cos(x)}{x}\]

The equality can be rewritten as:

\[ y’ = y \cdot \left(-2\sin(x) \cdot \ln(x) + \frac{2\cos(x)}{x} \right)\]

Since \(y = x^{2cosx}\), we have:

\[ y’ = x^{2cosx} \cdot \left(-2\sin(x) \cdot \ln(x) + \frac{2\cos(x)}{x} \right)\]