Compute the 6th derivative of function

Question

Compute the 6th derivative of f(x) = arctan⁡( (x^(2)) / 7)at x = 0.

(f^(6)) (0) = ?

Answer 1

There are two ways in which we could do this problem. The much longer method would be to differentiate f(x) six times and then evaluate at x = 0, which would entail several product, quotient, and chain rules!

Alternatively, we can use a Maclaurin series expansion for f(x) = arctan(x²/7).

Recall that arctan(u) = u-u³/3+u⁵/5-u⁷/7+..., where u is a function of x

Substitute u=x²/7 and we have arctan(x²/7) = x²/7-x⁶/(3*7³)+...

Note that by definition of a Maclaurin series, f⁽⁶⁾(0) / (6!) = the coefficient of the x⁶ term in the Maclaurin series of arctan(x²/7).

The coefficient on the x⁶ term is -1/(3*7³) = -1/1029.

So we have f⁽⁶⁾(0) / (6!) = -1/1029 and multiplying both sides by 6!, we get that f⁽⁶⁾(0) = -6! / 1029 = -720/1029 = -240/343.