Final answer:
To find dy/dx by implicit differentiation, apply the product rule and chain rule to both sides of the equation x*sin(y)*y*sin(x) = 6, differentiate with respect to x, and solve for dy/dx.
Step-by-step explanation:
The process of finding dy/dx by implicit differentiation for the equation x*sin(y)*y*sin(x) = 6 involves differentiating both sides of the equation with respect to x, remembering to apply the product rule and the chain rule for differentiation. As both x and y are functions of each other, we treat y as a function of x when applying the derivatives. This means that every time we differentiate a term with y in it, we multiply by dy/dx (also written as y').
Let's differentiate both sides of the equation:
The Differentiation Process:
1. Differentiate x with respect to x: d(x)/dx = 1.
2. Differentiate sin(y) with respect to x applying the chain rule: d(sin(y))/dx = cos(y)*dy/dx.
3. Apply the product rule to x*sin(y): d(x)*sin(y) + x*d(sin(y)).
4. Differentiate y*sin(x) similarly, treating y as a constant when differentiating sin(x) and using the product rule for y.
5. Set the derivative of the left side of the equation equal to the derivative of 6, which is 0.
6. Solve for dy/dx.
Note: Since this solution involves a sequence of algebraic steps that will include product rule and chain rule, it is advisable for the student to have knowledge of these differentiation rules to follow the process.