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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability2 Marks
Question
Differentiate the ${e^x} + {e^{{x^2}}} + ... + {e^{{x^5}}}$ w.r.t. x.
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Answer

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

$\therefore \frac{{dy}}{{dx}} = \frac{d}{{dx}}{e^x} + \frac{d}{{dx}}{e^{{x^2}}} + \frac{d}{{dx}}{e^{{x^3}}} + \frac{d}{{dx}}{e^{{x^4}}} + \frac{d}{{dx}}{e^{{x^5}}}$

$= {e^x} + {e^{{x^2}}}\frac{d}{{dx}}{x^2} + {e^{{x^3}}}\frac{d}{{dx}}{x^3} + {e^{{x^4}}} \frac{d}{dx}{x^4}+ {e^{{x^5}}}\frac{d}{{dx}}{x^5}$

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

$ = {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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