Tentukan d^4y/dx^4 jika y=x^(7/2)-2x^(5/2)+x^(1/2).

d^4y/dx^4 = (105/16)x^(-1/2) + (15/8)x^(-3/2) - (15/16)x^(-7/2)

Pembahasan

Untuk menentukan d^4y/dx^4 dari fungsi y=x^(7/2)-2x^(5/2)+x^(1/2), kita perlu melakukan diferensiasi sebanyak empat kali.

1. **Diferensiasi Pertama (dy/dx):**
   dy/dx = (7/2)x^(5/2) - 2*(5/2)x^(3/2) + (1/2)x^(-1/2)
   dy/dx = (7/2)x^(5/2) - 5x^(3/2) + (1/2)x^(-1/2)

2. **Diferensiasi Kedua (d^2y/dx^2):**
   d^2y/dx^2 = (7/2)*(5/2)x^(3/2) - 5*(3/2)x^(1/2) + (1/2)*(-1/2)x^(-3/2)
   d^2y/dx^2 = (35/4)x^(3/2) - (15/2)x^(1/2) - (1/4)x^(-3/2)

3. **Diferensiasi Ketiga (d^3y/dx^3):**
   d^3y/dx^3 = (35/4)*(3/2)x^(1/2) - (15/2)*(1/2)x^(-1/2) - (1/4)*(-3/2)x^(-5/2)
   d^3y/dx^3 = (105/8)x^(1/2) - (15/4)x^(-1/2) + (3/8)x^(-5/2)

4. **Diferensiasi Keempat (d^4y/dx^4):**
   d^4y/dx^4 = (105/8)*(1/2)x^(-1/2) - (15/4)*(-1/2)x^(-3/2) + (3/8)*(-5/2)x^(-7/2)
   d^4y/dx^4 = (105/16)x^(-1/2) + (15/8)x^(-3/2) - (15/16)x^(-7/2)

Jadi, d^4y/dx^4 = (105/16)x^(-1/2) + (15/8)x^(-3/2) - (15/16)x^(-7/2)