Find dy/dx for (x^2)(y^3) + ln(x^y) = 5sin(6x)/x^(1/2)

To differentiate this equation we must use the implicit method, looking at the left side of the equation, to find the differential of the first term we must use the product rule, therefore the differential of x²y³ is:
2xy³ + (x²3y²)(dy/dx)
For every y term that we differentiate, we must place a dy/dx behind it. For the next term we need to use a combination of the log laws and the product rule, so knowing that ln(x^y) = yln(x) we can see that the differential of ln(x^y) is:
(dy/dx)(ln(x)) + (y)(1/x)
To calculate the right side of the equation we must use the quotient rule, so the differential of 5sin(6x)/x^1/2 is:
(30cos(6x)x^1/2 - 5sin(6x)(1/2)x^-1/2)/x which simplifies to: (60xcos(6x) - 5sin(6x))/2x^3/2
So now we will simplify it and move like terms to either side:
(dy/dx)((x²3y²) + (ln(x))) = ((60xcos(6x) - 5sin(6x))/2x^3/2) - 2xy³ - y/x
So, the final answer is:
dy/dx = (((60xcos(6x) - 5sin(6x))/2x^3/2) - 2xy³ - y/x)/(x²3y² + (ln(x)))