Final answer:
To find the slope of a curve at a given point using implicit differentiation, we differentiate the equation with respect to x, solve for the derivative, and substitute the given point to find the slope.
Step-by-step explanation:
Implicit differentiation
To find the slope of the curve at the given point, we need to differentiate the equation implicitly. Let's say the equation of the curve is represented by y=f(x). To find the derivative dy/dx, we differentiate both sides of the equation with respect to x, treating y as a function of x.
Next, we solve the resulting equation for dy/dx to find the derivative. Once we have the derivative, we can substitute the x-coordinate of the given point to find the slope of the curve at that point.
Example:
Let's say the equation is x^2 + y^2 = 9 and we want to find the slope of the curve at the point (3, 0). The first step is to differentiate both sides with respect to x, treating y as a function of x:
2x + 2y(dy/dx) = 0
Now, we can solve this equation for dy/dx:
dy/dx = -x/y
Substituting the x-coordinate (3) and y-coordinate (0) of the given point into the derivative, we find that the slope at the point (3, 0) is 0.