Question

Differentiate the following w.r.t. $x$:
$e^x+e^{x^2}+....+e^{x^5}$

Answer 1

The correct answer is $=e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^3}$
Let $y=e^x+e^{x^2}+....+e^{x^5}$
$\frac{d}{dx}=\frac{d}{dx}[e^x+e^{x^2}+....+e^{x^5}]$
$=\frac{d}{dx}(e^x)+\frac{d}{dx}(e^{x^2})+\frac{d}{dx}(e^{x^4})+\frac{d}{dx}(e^{x^5})$
$=e^x+[e^{x^2}.\frac{d}{dx}(x^2)]+[e^{x^3}.\frac{d}{dx}(x^3)]+[e^{x^4}.\frac{d}{dx}(x^4)]+[e^{x^5}.\frac{d}{dx}(x^5)]$
$=e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^3}$