a₀ = 0, aₙ = 7/n⁴π, bₙ = 0.
1. Derivative of sin(nπt):
The derivative of sin(nπt) is nπ cos(nπt). This applies to each term in the summation of f(t).
2. Term-by-term differentiation:
Applying the derivative to each term of the series and keeping in mind the constant factor of 7/n⁵:
f'(t) = d/dt [∑ 7/n⁵ sin(nπt)]
= ∑ 7/n⁵ d/dt [sin(nπt)]
= ∑ 7/n⁵ * nπ cos(nπt)
3. Separating coefficients:
The resulting series has the same form as the desired answer with cos(nπt) for each term. We can directly compare coefficients to solve for aₙ and bₙ:
a₀: The constant term comes from differentiating any constant term in the original series, but there is none. Therefore, a₀ = 0.
aₙ: This comes from the coefficient of cos(nπt) in the differentiated series, which is 7/n⁵ * nπ. Therefore, aₙ = 7/n⁴π.
bₙ: Since only cosine terms appear after differentiation, there are no sine terms in the derivative series. Therefore, all bₙ are 0.
Therefore, the coefficients are:
a₀ = 0
aₙ = 7/n⁴π
bₙ = 0
This gives us the complete series for the derivative f'(t):
f'(t) = ∑ (7/n⁴π) * cos(nπt)