Final answer:
The slope of the tangent line to the curve y = sin(xy²) at the point (-1, -1) is cos(-x) * (1 - 2x * dy/dx).
Step-by-step explanation:
To find the slope of the tangent line to the curve y = sin(xy²) at the point (x, -1), we need to use the derivative. The derivative of sin(xy²) with respect to x can be found using the chain rule. Let's denote u=xy², then the derivative of sin(u) with respect to x is cos(u) * d(u)/dx = cos(xy²) * (y² + 2xy * dy/dx). Now, we can substitute the point (-1, -1) into the derivative expression to find the slope of the tangent line.
Substituting y = -1 into the expression, we get cos((-1)(x(-1)²)) * ((-1)² + 2(x)(-1) * dy/dx). Simplifying further, the expression becomes cos(-x) * (1 - 2x * dy/dx).
Therefore, the slope of the tangent line to the curve y = sin(xy²) at the point (-1, -1) is cos(-x) * (1 - 2x * dy/dx).