Let us consider this function:
f(x) = x13*cos2(x1000)*ex^3.
Determine f(2022)(0), that is, evaluating the 2022nd derivative of f(x) at x=0.
Note: Computer program simulations are certainly acceptable , but an semi-analytic method would be much better.
Differentiating the cos^2(x^1000) and e^(x^3) terms will increase the power on x^n by 999 and 2 respectively. Differentinating the x^n term will reduce the power on x^n by 1.
But cos^2(x^1000) needs to be differentiated twice to not be zero when evaluated at zero.
Also we need x^n to be differentiated down to a zero-degree constant.
The only way that I see for this to work is differentiate cos^2(x^1000) twice, and e^(x^3) three times. This makes the exponent x^n up to 13+1998+6=2017, which is exactly the remaining 2022-2-3=2017 differentiations.
But these differentiations can be interspersed amongst each other, and that is where I get lost in calculations.
Thoughts
