Differentiate the following w.r.t. x:
Question:

Differentiate the following w.r.t. x:

$e^{x}+e^{x^{2}}+\ldots+e^{x^{5}}$

 

Solution:

$\frac{d}{d x}\left(e^{x}+e^{x^{2}}+\ldots+e^{x^{5}}\right)$

$=\frac{d}{d x}\left(e^{x}\right)+\frac{d}{d x}\left(e^{x^{2}}\right)+\frac{d}{d x}\left(e^{x^{3}}\right)+\frac{d}{d x}\left(e^{x^{4}}\right)+\frac{d}{d x}\left(e^{x^{3}}\right)$

$=e^{x}+\left[e^{x^{2}} \times \frac{d}{d x}\left(x^{2}\right)\right]+\left[e^{x^{3}} \cdot \frac{d}{d x}\left(x^{3}\right)\right]+\left[e^{x^{4}} \cdot \frac{d}{d x}\left(x^{4}\right)\right]+\left[e^{x^{5}} \cdot \frac{d}{d x}\left(x^{5}\right)\right]$

$=e^{x}+\left(e^{x^{2}} \times 2 x\right)+\left(e^{x^{3}} \times 3 x^{2}\right)+\left(e^{x^{4}} \times 4 x^{3}\right)+\left(e^{x^{3}} \times 5 x^{4}\right)$

$=e^{x}+2 x e^{x^{2}}+3 x^{2} e^{x^{3}}+4 x^{3} e^{x^{4}}+5 x^{4} e^{x^{5}}$

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