To differentiate $y = x^{\sin{x}}$, we can use logarithmic differentiation. Here's the process:

1. Take the natural logarithm of both sides to make the exponent easier to handle: $\ln{y} = \ln{(x^{\sin{x}})}$ Using the property of logarithms $\ln{(a^b)} = b\ln{a}$, this becomes: $\ln{y} = \sin{x} \cdot \ln{x}$

2. Differentiate both sides with respect to $x$ (using implicit differentiation on the left and the product rule on the right): $\frac{1}{y} \cdot \frac{dy}{dx} = \cos{x} \cdot \ln{x} + \sin{x} \cdot \frac{1}{x}$

3. Solve for $\frac{dy}{dx}$ by multiplying both sides by $y$: $\frac{dy}{dx} = y \left( \cos{x} \cdot \ln{x} + \frac{\sin{x}}{x} \right)$

4. Substitute back $y = x^{\sin{x}}$: $\frac{dy}{dx} = x^{\sin{x}} \left( \cos{x} \cdot \ln{x} + \frac{\sin{x}}{x} \right)$

Thus, the derivative of $y = x^{\sin{x}}$ is: $\frac{dy}{dx} = x^{\sin{x}} \left( \cos{x} \cdot \ln{x} + \frac{\sin{x}}{x} \right)$

Would you like any additional details or further explanation?

Here are 5 related questions:

1. How would you find the second derivative of $y = x^{\sin{x}}$?
2. Can logarithmic differentiation be used for other types of functions? If so, how?
3. What is the domain of $x^{\sin{x}}$ for real values of $x$?
4. How does the derivative behave near $x = 0$?
5. How can we extend this to functions of the form $x^{f(x)}$?

Tip: When faced with an exponent involving a variable, logarithmic differentiation is a powerful tool to simplify differentiation.