find the derivative of ∑ (k - 3)x^(k + 2)/(k - 2)
find the derivative of ∑ (k - 3)x^(k + 2)/(k - 2)
Answer
Explanation:
Step1: Recall power - rule for differentiation
The power - rule states that if (y = x^n), then (y^\prime=nx^{n - 1}). Also, if (y=\sum_{k}a_kx^{n_k}), then (y^\prime=\sum_{k}a_kn_kx^{n_k - 1}). Here (a_k=\frac{k - 3}{k - 2}) and (n_k=k + 2).
Step2: Apply the power - rule to the series
Differentiating term - by - term, we get (\sum_{k}\frac{k - 3}{k - 2}(k + 2)x^{(k + 2)-1}=\sum_{k}\frac{(k - 3)(k + 2)}{k - 2}x^{k+1}).
Answer:
(\sum_{k}\frac{(k - 3)(k + 2)}{k - 2}x^{k + 1})