Question

$\mathtt \purple{if \: y = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } \: \: find \: \: \frac{dy}{dx}. }$
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Answer 1

Answer:

y - x^n /n!

Step-by-step explanation:

⇒ y = 1 + x + x²/2! + x³/3! + ... + xⁿ/n!

          Differentiate w.r.t x

⇒ y' = d(1 + x + x²/2! + x³/3! + ... + xⁿ/n!)/dx

       = $\frac{d}{dx} 1 + \frac{d}{dx}\frac{x}{1 \ !} + \frac{d}{dx}\frac{x^2}{2 \ !}+ \frac{d}{dx}\frac{x^3}{3 \ !} + ... + \frac{d}{dx}\frac{x^n}{n \ !}$

       = $\frac{d}{dx} 1 + \frac{d}{dx}x + \frac{1}{2 \ !} \frac{d}{dx} x^2+ \frac{1}{3 \ !}\frac{d}{dx}x^3+ ... + \frac{1}{n \ !}\frac{d}{dx}x^n$

       = $0 + 1 + \frac{1}{2 \ !} (2x)+ \frac{1}{3 \ !}(3x^2)+ ... + \frac{1}{n \ !} (nx^{n-1})$

       = $1 + \frac{1}{2\times 1 \ !} (2x)+ \frac{1}{3 \times 2 \ !}(3x^2)+ ... + \frac{1}{n \times (n-1) \ !} (nx^{n-1})$

       = $1 + \frac{1}{1 \ !} x + \frac{1}{2 \ !}(x^2)+ ... + \frac{1}{(n-1) \ !} (x^{n-1})$

       = $1 + x + \frac{x^2}{2 \ !}+ ... + \frac{x^{n-1}}{(n-1) \ !}$

       = $1 + x + \frac{x^2}{2 \ !}+ ... + \frac{x^{n-1}}{(n-1) \ !} +\frac{x^n}{n \ !}- \frac{x^n}{n \ !}$

       = $y - \frac{x^n}{n \ !}$

Therefore,

   $\frac{dy}{dx} = y - \frac{x^n}{n \ !}$

Using: n! = n * (n - 1)!

      d(x^n)/dx = n * x^(n-1)

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Answer 2

Answer:

$\qquad\qquad\underline{\textsf{\textbf{ \color{green}{Solution\:completed} }}}$

Question :-

$\sf \pink{if \: y = 1 + x + \dfrac { x ^ { 2 } } { 2 ! } + \dfrac { x ^ { 3 } } { 3 ! } + \cdots + \dfrac { x ^ { n } } { n ! } \: \: find \: \: \dfrac{dy}{dx}. }$

Given :-

➳ $\sf if \: y = 1 + x + \dfrac { x ^ { 2 } } { 2 ! } + \dfrac { x ^ { 3 } } { 3 ! } + \cdots + \dfrac { x ^ { n }}{ n ! }$.

Find Out :-

➳ $\sf \dfrac{dy}{dx}$.

Solution :-

✬ $\red{ \boxed{\sf{\dfrac{dy}{dx} =\: y - \dfrac{x^n}{n!}}}}$ ✬

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Step-by-step explanation:

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