Question

what is the 100th derivative of f(x)=x^3*e^x

Answer 1

Oh, that looks fun. What have you tried? Did you calculate the first four?

Answer 2

i did
not clear yet

Answer 3

I found up to the first 7. I found that there is a pattern for all four terms but I can not figure out the generalization for the terms yet

Answer 4

so far I have figured out that f^n(x)= x^3*e^x +3n*x^2*e^x +_____x*e^x+____e^x

Answer 5

the ____ portions represent the generalizations I have not gotten yet

Answer 6

for the x*e^x term the coeefficients are (in order of first derivative to 6th): 0, 6, 18, 36, 60, 90

Answer 7

for the e^x term the coefficients are : 0, 0, 6, 24, 60, 120

Answer 8

??

Answer 9

You have it. You're just giving up too soon.

Given \(e^{x}(A_{n}x^{3} + B_{n}x^{2} + C_{n}x + D_{n})\), we have:

\(A_{n} = 1\)
\(B_{n} = 3n\) for \(n \ge 0\) and 0 otherwise.
\(C_{n} = 3n(n-1)\) for \(n \ge 1\) and 0 otherwise.
\(D_{n} = n(n-1)(n-2)\) for \(n \ge 2\) and 0 otherwise.

You have the \(D_{n}\) exactly correct. What's stopping you from writing the appropriate cubic expression to reproduce it?

Now prove it!

Answer 10

I've been teaching math for the last 5 years and haven't worked with calculus in 9 years... a little too out of practice and I don't have any of my books with me
thanks!

Answer 11

You seem to have the calculus of this problem just fine. It's the other stuff that stumped you. Recognizing the pattern of Constant, Linear, Quadratic, Cubic and building the separate functions requires no calculus.

Get up to speed. You do NOT want to miss THAT student when the opportunity presents itself.