1. The problem asks to find the derivative with respect to $x$ of the expression $x^{1+x}$.\n\n2. The function is $y = x^{1+x}$. To differentiate this, we use logarithmic differentiation because the exponent itself depends on $x$.\n\n3. Take the natural logarithm of both sides: $$\ln y = \ln \left(x^{1+x}\right) = (1+x) \ln x.$$\n\n4. Differentiate both sides with respect to $x$ using implicit differentiation: $$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[(1+x) \ln x].$$\n\n5. Use the product rule on the right side: $$\frac{d}{dx}[(1+x) \ln x] = (1+x) \frac{1}{x} + \ln x \cdot 1 = \frac{1+x}{x} + \ln x.$$\n\n6. Multiply both sides by $y$ to solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = y \left(\frac{1+x}{x} + \ln x\right).$$\n\n7. Substitute back $y = x^{1+x}$: $$\frac{dy}{dx} = x^{1+x} \left(\frac{1+x}{x} + \ln x\right).$$\n\n8. Simplify the expression inside the parentheses: $$\frac{1+x}{x} = \frac{1}{x} + 1,$$ so $$\frac{dy}{dx} = x^{1+x} \left(1 + \frac{1}{x} + \ln x\right).$$\n\n9. None of the options a, b, c, or d exactly match this derivative. Therefore, the correct answer is option e: None of the given options is correct.