You know that f(x) can be expanded as a series as follows.
f(x) = f(0) + f'(0)*x + (f''(0)/2)*x^{2 }+ (f'''(0)/6)*x^{3} + (f''''(0)/24)*x^{4} + ...
In case of cos(x), the coefficient of any odd power is zero, which matches well with the fact that cos(x) and even power are even functions.
cos(x) = 1 1/2*x^{2} + 1/24*x^{4} + ...
When we apply the chain rule in computing derivatives of cos(3x), the Maclaurin series for cos(3x) is computed as follows,
cos(3x) = 1 (3^{2}/2)*x^{2} + (3^{4}/24)*x^{4} + ....
So, the coefficient of x^{4} is 3^{4}/24, which is 27/8.
The answer: a) 27/8
1/26/2014

Isaac K.
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