d/dx ((x^2 akar(x))^(1/3)-(x x^(1/3))^(1/5))=....

(5/6)x^(-1/6) - (4/15)x^(-11/15)

Pembahasan

Untuk menghitung turunan dari fungsi yang diberikan, kita perlu menggunakan aturan turunan dasar dan aturan rantai.

Pertama, mari kita sederhanakan ekspresi di dalam kurung:

x^2 * akar(x) = x^2 * x^(1/2) = x^(2 + 1/2) = x^(5/2)
Jadi, (x^2 akar(x))^(1/3) = (x^(5/2))^(1/3) = x^(5/6)

x * x^(1/3) = x^(1 + 1/3) = x^(4/3)
Jadi, (x x^(1/3))^(1/5) = (x^(4/3))^(1/5) = x^(4/15)

Sekarang, fungsi yang perlu diturunkan adalah: f(x) = x^(5/6) - x^(4/15)

Menggunakan aturan turunan d/dx (x^n) = n*x^(n-1):

d/dx (x^(5/6)) = (5/6) * x^((5/6) - 1) = (5/6) * x^(-1/6)

d/dx (x^(4/15)) = (4/15) * x^((4/15) - 1) = (4/15) * x^(-11/15)

Maka, d/dx ((x^2 akar(x))^(1/3)-(x x^(1/3))^(1/5)) = (5/6) * x^(-1/6) - (4/15) * x^(-11/15)

Jawaban ringkasnya adalah: (5/6)x^(-1/6) - (4/15)x^(-11/15)