Is there an order of operations to follow when finding derivatives

No not pemdas....
What I mean is; when Finding a derivative of an equation, is there a certain order to follow
For example: f= 6x+ xsin^2(x^2y)
In which order should I use the rules in order to find the derivative?

Well derivative of 6x is simple enough:
d/dx (6x) = 6

For x sin²(x²y), use
product rule (since we have 2 functions: x and sin²(x²y)
power rule/chain rule on sin²(x²y)
chain rule on sin(x²y)
product rule on (x²y)

Notice how we always go from outer functions to inner functions.
So order of differentiation has nothing to do with what kind of rule you use (product rule, chain rule, quotient rule) but on the the nesting of the functions, always working your way from outer functions to inner functions.

d/dx (x sin²(x²y))
= d/dx (x) * sin²(x²y) + x * d/dx (sin²(x²y)) . . . . . . . . . product rule
= sin²(x²y) + x * 2 sin(x²y) * d/dx (sin(x²y)) . . . . . . . . chain rule
= sin²(x²y) + 2x sin(x²y) * cos(x²y) * d/dx (x²y) . . . . . chain rule
= sin²(x²y) + 2x sin(x²y) cos(x²y) * (d/dx (x²) * y + x² * d/dx (y)) . . . . . product rule
= sin²(x²y) + 2x sin(x²y) cos(x²y) (2xy + x² y')

Mαthmφm

You mean like the chain rule? Other than that it really shouldn't matter...

6x + xsin^2(x^2y)
6 + [1*sin^2(x^2y) + x*2sin(x^2y)] * 2sin(x^2y) * (2xy + x^2*dy/dx)

Now isolate dy/dx.