Question

Use the MacLaurin series for f(x).

Compute the 10th derivative of
f(x)=arctan(x^2/4)
at x=0.
f^(10)(0)=

Answer 1

y = tan^−1(x2/4) tan(y) = x2/4 sec2(y) = x/2 y′ = xcos^2(y)/2 cos^2(y) = 16x2+16 y′ = 8x/(x2+16)

let u be x2+16
du is 2x dx
dy = 4 du / u
y = 4 ln (x2 + 16)

y at x =0 = 4 ln (16) = 11.09