${e^x} + {e^{{x^2}}} + ..... + {e^{{x^5}}}$
${e^x} + {e^{{x^2}}} + ..... + {e^{{x^5}}}$
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Let $y = {e^x} + {e^{{x^2}}} + ..... + {e^{{x^5}}}$
therefore, $\cfrac{{dy}}{{dx}} = \cfrac{d}{{dx}}({e^x} + {e^{{x^2}}} + ..... + {e^{{x^5}}})$
$= \cfrac{d}{{dx}}({e^x}) + \cfrac{d}{{dx}}({e^{{x^2}}}) + \cfrac{d}{{dx}}({e^{{x^3}}}) + \cfrac{d}{{dx}}({e^{{x^4}}}) + \cfrac{d}{{dx}}({e^{{x^5}}})$
$= {e^x} + {e^{{x^2}}}(2x) + {e^{{x^3}}}(3{x^2}) + {e^{{x^4}}}(4{x^3}) + {e^{{x^5}}}(5{x^4})$
$= {e^x} + 2x{e^{{x^2}}} + 3{x^2}{e^{{x^3}}} + 4{x^3}{e^{{x^4}}} + 5{x^4}{e^{{x^5}}}$
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