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Problem 63 Hard Difficulty

Find expressions for the first five derivatives of $ f(x) = x^2e^x. $ Do you see a pattern in these expressions? Guess a formula for $ f^{(n)}(x) $ and prove it using mathematical induction.

Answer

1. $S_{1}$ is true because $f^{\prime}(x)=e^{x}\left(x^{2}+2 x\right)$
2. $=e^{x}\left[x^{2}+2(k+1) x+(k+1) k\right]$
3. $f^{(n)}(x)=e^{x}\left[x^{2}+2 n x+n(n-1)\right]$

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Adam R.

March 5, 2019

Video Transcript

you know it's clear. So when you raid here, so we have f of X is equal to x square times Ito X. When we differentiate for the first derivative, we apply the product rule and we get eat the X times X square plus two x For a second derivative, we get eat to the axe X square plus four X plus two. And for the third derivative, we eat the X X squared plus six X plus six. For the fourth derivative we could eat the X comes X square plus eight X plus 12. And for the 50 riveted, we got beat to the ex X Square plus 10 X plus 20. And if we see here, we see that the second coefficient is a function of end. So we get our function, which is eat the x X square close to an ex plus and times and minus one. Now, to prove this, we're going to make first unequal one. So this equals just the derivative, the first derivative. So one equation is satisfied. Now we're gonna look for n as equal to K, and we get f is equal to eat the axe times X square plus two k x plus K times came minus one. And finally we're going to proven as equal to K plus one. And this is equal Thio. Hey, the derivative which is equal to eat The X Times X squared plus two k x plus k times came minus one plus e to the X terms to X plus two k Ms is equal to eat to the ex times X square close two times K plus one ex plus que plus one cake So we have proved our formula.