Find the first three non-zero terms of the Taylor series for f(x) = tan(x).

We have that the Taylor series of a function infinitely differentiable at a x = a is given by the expansion: f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + f⁽⁴⁾(a)(x - a)⁴​​​​​​​/4! +... Thus we differentiate f(x) 5 times and evaluate at zero (as in this case a = 0) in order to obtain all our coefficients. f(x) = tan(x), f(0) = tan(0) = 0 f'(x) = sec²(x) = 1 + tan²(x) = 1 + f(x)², thus f'(0) = 1 + f(0)² = 1 [by writing f'(x) in terms of f(x), we can skip differentiating reciprocal trig functions and simply leave the derivates in terms of f(x) and its derivatives of lower order] f''(x) = 2f'(x)f(x), f''(0) = 0 f'''(x) = 2(f''(x)f(x) + f'(x)²), f'''(0) = 2 f⁽⁴⁾(x) = 2(f'''(x)f(x) + 3f''(x)f'(x)), f⁽⁴⁾(0) = 0 f⁽⁵⁾(x) = 2(f⁽⁴⁾(x)f(x) + 4f'''(x)f'(x) + 3f''(x)²), f⁽⁵⁾(0) = 16 Thus the first three non-zero terms of the Taylor series for tan(x) are: x + 2x³/3! + 16x⁵/5! = x + x³/3 + 2x⁵/15