Help with calculus

Find the n th derivative of y, if
y = (1-x^2)cosx

So we're asked for an explicit formula for the n th drivative. Thanks

Question

Help with calculus

Find the n th derivative of y, if
y = (1-x^2)cosx

So we're asked for an explicit formula for the n th drivative. Thanks

science0229 · Answer

These kind of problem always has a pattern.
I would keep differentiating it until there is a pattern.

anonymous · Answer

I can get the pattern for cosx, the problem is that x^2cosx, the derivative just gets larger end larger

science0229 · Answer

$$f(x)=(1-x^2)cosx$$
$$f'(x)=(cosx-x^2cosx)'=-sinx-2xcosx+x^2sinx=-2xcosx-(1-x^2)sinx$$
$$f''(x)=(-2xcosx-sinx+x^2sinx)'=-2cosx+2xsinx-cosx+2xsinx+$$
$$x^2cosx$$

science0229 · Answer

The second derivative simplifies farther to $$4xsinx-(3-x^2)cosx$$

science0229 · Answer

The third derivative skipping the calculation, is $$6xcosx+(7-x^2)sinx$$

science0229 · Answer

The fourth derivative is $$-8xsinx+(13-x^2)cosx$$

science0229 · Answer

The fifth derivative is $$-10xcosx-(21-x^2)sinx$$

science0229 · Answer

See a pattern?

anonymous · Answer

Maybe f^n(x) = 2nxsin(x+n*pi/2)+(somenumber-x^2)cos(x+n*pi/2)

anonymous · Answer

but what is that number

science0229 · Answer

that "somenumber" are 1,3,7,13,21...etc

1+2=3
3+4=7
7+6=13
13+8=21
...

anonymous · Answer

ok, i got it, thanks

science0229 · Answer

Welcome:)