x^2 sin⁡(xy)+y = x

Tentukan turunan pertama dari bentuk implisit x^2 sin⁡(xy)+y=x !

Jawab :

Turunan bentuk implisit

Pertama kita turunkan dulu sin ⁡(xy)

Misalkan : u = xy

maka: du/dx=(1)(y)+ (x)(dy/dx)=y+x dy/dx

Maka: y = sin u maka: dy/du=cos⁡u=cos⁡(xy)

Sehingga :

dy/dx=(dy/du)(du/dx)

dy/dx=(cos⁡(xy) )(y+x dy/dx)

dy/dx=y cos⁡(xy)+ x cos⁡(xy) dy/dx

 

Kedua kita turunkan x^2 sin⁡(xy)

d(x^2 sin⁡(xy)/dx = (2x)(sin⁡(xy)) + (x^2)[y cos⁡(xy)+ x cos⁡(xy) dy/dx]

d(x^2 sin⁡(xy)/dx = 2x sin⁡(xy) + x^2 y cos⁡(xy) + x^3cos⁡(xy) dy/dx

Maka:

d(x^2 sin⁡(xy)+y)/dx = d(x)/dx

2x sin⁡(xy) + x^2 y cos⁡(xy) + x^3cos⁡(xy) dy/dx + d(y)/dx = 1

dy/dx [x^3cos⁡(xy) + 1] = 1 – 2x sin⁡(xy) -x^2 y cos⁡(xy)

dy/dx =[1 – 2x sin⁡(xy) – x^2 y cos⁡(xy)] / [x^3cos⁡(xy) + 1]