Differentiate the following with respect to $x$: $e^{x}+e^{x^{2}}+e^{x^{3}}+e^{x^{4}}+e^{x^{5}}$

Let $y = e^{x}+e^{x^{2}}+e^{x^{3}}+e^{x^{4}}+e^{x^{5}}$.
To find the derivative with respect to $x$,we apply the sum rule and the chain rule:
$\frac{dy}{dx} = \frac{d}{dx}(e^{x}) + \frac{d}{dx}(e^{x^{2}}) + \frac{d}{dx}(e^{x^{3}}) + \frac{d}{dx}(e^{x^{4}}) + \frac{d}{dx}(e^{x^{5}})$
Using the chain rule $\frac{d}{dx}(e^{u}) = e^{u} \cdot \frac{du}{dx}$:
$= e^{x} + e^{x^{2}} \cdot \frac{d}{dx}(x^{2}) + e^{x^{3}} \cdot \frac{d}{dx}(x^{3}) + e^{x^{4}} \cdot \frac{d}{dx}(x^{4}) + e^{x^{5}} \cdot \frac{d}{dx}(x^{5})$
$= e^{x} + e^{x^{2}} \cdot (2x) + e^{x^{3}} \cdot (3x^{2}) + e^{x^{4}} \cdot (4x^{3}) + e^{x^{5}} \cdot (5x^{4})$
$= e^{x} + 2x e^{x^{2}} + 3x^{2} e^{x^{3}} + 4x^{3} e^{x^{4}} + 5x^{4} e^{x^{5}}$