Question

The point (3,1) lies on the curve in the xy-plane given by the equation (f(x))² + g(x) = xy -1. What is the value of dy/dx at the point?

Answer 1

The value of dy/dx at the point (3,1) can be found by implicitly differentiating the given curve equation, substituting the point into the differentiated equation, and solving for dy/dx. Additional information about f(x) and g(x) is required.

Step-by-step explanation:

To find the value of dy/dx at the point (3,1) given the curve equation (f(x))² + g(x) = xy - 1, we first need to apply implicit differentiation to the equation with respect to x. Differentiate each term separately, remembering to use the chain rule on the (f(x))² term. The given point (3,1) means that when x is 3, the y-value is 1.

Start by differentiating both sides with respect to x:

1. For (f(x))², the derivative is 2f(x)f'(x).
2. For g(x), the derivative is simply g'(x).
3. On the right side, the derivative of xy is y + x(dy/dx), using the product rule.
4. And the derivative of a constant (-1) is 0.

So, the differentiated equation becomes 2f(x)f'(x) + g'(x) = y + x(dy/dx).

Substituting the point (3,1) into this new equation, you can solve for dy/dx by using the values of f(x), f'(x), and g'(x) at x=3. This generally requires additional information about the functions f(x) and g(x), which is not provided here. If these derivative functions are known, direct substitution will yield the specific value for dy/dx.