From the uploaded images, I can see the problem involves finding the derivative of the function:

$f(x) = x^{2x}$

To differentiate this function, we will use logarithmic differentiation.

Steps:

1. Take the natural logarithm (ln) on both sides: $\ln f(x) = \ln \left( x^{2x} \right)$

2. Simplify using the logarithmic property $\ln(a^b) = b \ln a$: $\ln f(x) = 2x \ln x$

3. Differentiate both sides with respect to $x$, remembering that $f(x)$ depends on $x$: $\frac{1}{f(x)} \frac{d}{dx} f(x) = \frac{d}{dx}(2x \ln x)$

4. Compute the derivative of $2x \ln x$ using the product rule: $\frac{d}{dx}(2x \ln x) = 2 \cdot \frac{d}{dx}(x \ln x) = 2 \left( \ln x + 1 \right)$

5. Substitute into the equation: $\frac{1}{f(x)} \frac{d}{dx} f(x) = 2 (\ln x + 1)$

6. Solve for $\frac{d}{dx} f(x)$: $\frac{d}{dx} f(x) = f(x) \cdot 2 (\ln x + 1)$

7. Replace $f(x)$ with $x^{2x}$: $\frac{d}{dx} f(x) = x^{2x} \cdot 2 (\ln x + 1)$

Final Answer:

$\frac{d}{dx} \big(x^{2x}\big) = x^{2x} \cdot 2 (\ln x + 1)$

Would you like further clarification or a step-by-step walkthrough?

Here are some related questions for deeper understanding:

1. What is logarithmic differentiation, and why is it useful here?
2. How does the product rule apply to the derivative of $x \ln x$?
3. Could the derivative be verified using numerical approximations?
4. How would this differ if the base of the power were not $x$?
5. How does this derivative behave for small or large values of $x$?

Tip: Use logarithmic differentiation whenever the variable is both in the base and the exponent, as it simplifies the process significantly!