Turunan kedua dari f(x)=sin⁴(¶/4 -x)

Turunan kedua dari f(x) = sin⁴ (π/4 - x)

Jawaban

Pendahuluan

Jika u ada suatu fungsi dalam x

y = sin u ⇒ y' = cos u . u'

y = cos u ⇒ y' = - sin u . u'

y = uⁿ ⇒ y' = n uⁿ⁻¹ . u'

y = u . v ⇒ y' = u' v + v' u

Rumus sudut rangkap pada sinus

2 sin x cos x = sin 2x

Pembahasan

f(x) = sin⁴ (π/4 - x)

f(x) = [sin (π/4 - x)]⁴

f'(x) = 4 [sin (π/4 - x)]³ . cos (π/4 - x) . -1

f'(x) = - 4 sin³ (π/4 - x) . cos (π/4 - x)

f'(x) = u . v

Misal

u = - 4 sin³ (π/4 - x) = -4 [sin (π/4 - x)]³

u' = -12 [sin (π/4 - x)]² . cos (π/4 - x) . -1

u' = 12 sin² (π/4 - x) cos (π/4 - x)

v = cos (π/4 - x)

v' = - sin (π/4 - x) . -1

v' = sin (π/4 - x)

f''(x) = u' v + v' u

f"(x) = 12 sin² (π/4 - x) cos (π/4 - x) . cos (π/4 - x) + sin (π/4 - x) . - 4 sin³ (π/4 - x)

f"(x) = 12 sin² (π/4 - x) cos² (π/4 - x) - 4 sin⁴ (π/4 - x)

Jika lebih sederhana lagi, maka kita ubah lagi

f"(x) = 3 . 4 sin² (π/4 - x) cos² (π/4 - x) - 4 sin⁴ (π/4 - x)

f"(x) = 3 . [2 sin (π/4 - x) cos (π/4 - x)]² - 4 sin⁴ (π/4 - x)

f"(x) = 3 . [sin 2(π/4 - x)]² - 4 sin⁴ (π/4 - x)

f"(x) = 3 . [sin (π/2 - 2x)]² - 4 sin⁴ (π/4 - x)

f"(x) = 3 sin² (π/2 - 2x) - 4 sin⁴ (π/4 - x)

Kesimpulan

f(x) = sin⁴ (π/4 - x)

maka turunan keduanya

f"(x) = 12 sin² (π/4 - x) cos² (π/4 - x) - 4 sin⁴ (π/4 - x)

f"(x) = 3 sin² (π/2 - 2x) - 4 sin⁴ (π/4 - x)

Pelajari lebih lanjut  

https://brainly.co.id/tugas/9759954

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Detil Jawaban  

Kelas : 12

Mapel : Matematika Peminatan

Kategori : Turunan

Kode : 12.2.2

Kata Kunci : Turunan trigonometri