a) 𝑦′ = 12𝗑2 + 2𝗑 − 𝑐𝑜𝑠𝗑
𝑦′′ = 24𝗑 + 2 + 𝑠𝑒𝑛𝗑
b) 𝑦 = 2𝑙𝑛𝗑 1
′ −1
𝑦 = 2 ( ) = 2𝗑
𝗑
𝑦′′ = −2𝗑−2
, Ejemplo 2
1 = 𝑎−1
Determine 𝑦(𝑛) donde 𝑦 = 1
𝑥+3 𝑎
Resolución
1) Hacer 𝑦 = (𝑥 + 3)−1 [𝑢𝑛]′ = 𝑛𝑢𝑛−1. 𝑢′
2) Derivar: 𝑦′ = −1(𝑥 + 3)−2. (𝑥 + 3)′ = −1(𝑥 + 3)−2
3)
𝑦(2) = 1.2(𝑥 + 3)−3. (𝑥 + 3)′ = 1.2(𝑥 + 3)−3 = 2! (𝑥 + 3)−(2+1)
𝑦(3) = −1.2.3(𝑥 + 3)−4 = −3! (𝑥 + 3)−(3+1)
𝑦(4) = 1.2.3.4(𝑥 + 3)−5 = 4! (𝑥 + 3)−(4+1)
𝑦(𝑛) = (−1)𝑛 𝑛! (𝑥 + 3)−(𝑛+1)
𝑦′′ = 24𝗑 + 2 + 𝑠𝑒𝑛𝗑
b) 𝑦 = 2𝑙𝑛𝗑 1
′ −1
𝑦 = 2 ( ) = 2𝗑
𝗑
𝑦′′ = −2𝗑−2
, Ejemplo 2
1 = 𝑎−1
Determine 𝑦(𝑛) donde 𝑦 = 1
𝑥+3 𝑎
Resolución
1) Hacer 𝑦 = (𝑥 + 3)−1 [𝑢𝑛]′ = 𝑛𝑢𝑛−1. 𝑢′
2) Derivar: 𝑦′ = −1(𝑥 + 3)−2. (𝑥 + 3)′ = −1(𝑥 + 3)−2
3)
𝑦(2) = 1.2(𝑥 + 3)−3. (𝑥 + 3)′ = 1.2(𝑥 + 3)−3 = 2! (𝑥 + 3)−(2+1)
𝑦(3) = −1.2.3(𝑥 + 3)−4 = −3! (𝑥 + 3)−(3+1)
𝑦(4) = 1.2.3.4(𝑥 + 3)−5 = 4! (𝑥 + 3)−(4+1)
𝑦(𝑛) = (−1)𝑛 𝑛! (𝑥 + 3)−(𝑛+1)
