Tentukan pertama dari fungsi turunan pecahan trigonometri

Turunan dari h(theta) adalah ((sec theta tan theta + cosec 2theta - 2 theta cosec 2theta cotan 2theta)(theta^2 sin theta - cotan theta) - (sec theta + theta cosec 2theta)(2 theta sin theta + theta^2 cos theta + cosec^2 theta)) / (theta^2 sin theta - cotan theta)^2

Pembahasan

Untuk menentukan turunan dari fungsi trigonometri h(theta)=(sec theta+ theta cosec 2theta)/(theta^2 sin theta-cotan theta), kita perlu menggunakan aturan kuosien dan aturan perkalian serta turunan dari fungsi-fungsi trigonometri dasar.

Misalkan u = sec theta + theta cosec 2theta dan v = theta^2 sin theta - cotan theta.

Mencari turunan u (u'):
d(sec theta)/d(theta) = sec theta tan theta
d(theta cosec 2theta)/d(theta) = (1 * cosec 2theta) + (theta * (-cosec 2theta cotan 2theta * 2))
= cosec 2theta - 2 theta cosec 2theta cotan 2theta
Jadi, u' = sec theta tan theta + cosec 2theta - 2 theta cosec 2theta cotan 2theta

Mencari turunan v (v'):
d(theta^2 sin theta)/d(theta) = (2 theta * sin theta) + (theta^2 * cos theta)
= 2 theta sin theta + theta^2 cos theta
d(cotan theta)/d(theta) = -cosec^2 theta
Jadi, v' = 2 theta sin theta + theta^2 cos theta - (-cosec^2 theta)
= 2 theta sin theta + theta^2 cos theta + cosec^2 theta

Menggunakan aturan kuosien (h'(theta) = (u'v - uv') / v^2):
h'(theta) = ((sec theta tan theta + cosec 2theta - 2 theta cosec 2theta cotan 2theta)(theta^2 sin theta - cotan theta) - (sec theta + theta cosec 2theta)(2 theta sin theta + theta^2 cos theta + cosec^2 theta)) / (theta^2 sin theta - cotan theta)^2

Jawaban Singkat: Turunan dari h(theta) adalah ((sec theta tan theta + cosec 2theta - 2 theta cosec 2theta cotan 2theta)(theta^2 sin theta - cotan theta) - (sec theta + theta cosec 2theta)(2 theta sin theta + theta^2 cos theta + cosec^2 theta)) / (theta^2 sin theta - cotan theta)^2