Final answer:
The slope of the tangent line to the curve can be found by differentiating the given equation with respect to x, and then solving for dy/dx. The value of dy/dx at the point (-1, π/6) gives the slope of the tangent.
Step-by-step explanation:
To find the slope of the tangent line to the curve at a given point, we need to take the derivative of the curve with respect to x and evaluate it at the point of interest. The given curve is 3x² y cosy=(√3 π)/2, and we need to find the tangent at the point (-1,π/6). We can use implicit differentiation to find dy/dx as the slope of the tangent line.
First, differentiate both sides of the equation with respect to x:
- Derive 3x² y with respect to x using the product rule.
- Derive cosy with respect to x considering y as a function of x.
- On the right side, since we have a constant, its derivative will be 0.
After differentiating, isolate dy/dx and evaluate it at the given point.
If the differentiation is done correctly, you would find that at the point (-1, π/6), the slope of the tangent line (dy/dx) can then be calculated by substituting the x and y values into this derivative.