Final Answer:
The derivative of the family of curves sin(xy) + x³ + y⁵= c with respect to x is given by dy/dx = (-cos(xy) - 3x²)/(y⁴ + x*cos(xy)).
Explanation:
To find the derivative of the given family of curves, we use implicit differentiation. We start by differentiating each term with respect to x. The derivative of sin(xy) with respect to x is cos(xy) (y + x dy/dx), the derivative of x^3 is 3x^2, and the derivative of y⁵ is 5y⁴ dy/dx. Then, we rearrange the terms to solve for dy/dx. After simplifying, we get the derivative dy/dx = (-cos(xy) - 3x²)/(y⁴ + xcos(xy)).
This process involves applying the chain rule and product rule for differentiation. The chain rule is used when differentiating sin(xy), and the product rule is used when differentiating xy. By isolating dy/dx, we obtain the final expression for the derivative of the family of curves with respect to x.
Implicit differentiation is a powerful technique used to find derivatives of equations that cannot be easily solved for y in terms of x. It allows us to find the rate of change of one variable with respect to another in implicit functions, such as the family of curves represented by sin(xy) + x³ + y⁵ = c.