58–59. Carry out the following steps.
b. Find the slope of the curve at the given point.
xy^5/2+x^3/2y=12; (4, 1)

Verified step by step guidance

1. First, understand that the problem requires finding the slope of the curve at a specific point, which involves calculating the derivative of the given equation implicitly.
2. The given equation is \( xy^{\frac{5}{2}} + x^{\frac{3}{2}}y = 12 \). To find the derivative, we will use implicit differentiation since the equation involves both x and y.
3. Differentiate both sides of the equation with respect to x. Remember to apply the product rule where necessary. For example, when differentiating \( xy^{\frac{5}{2}} \), treat it as a product of x and \( y^{\frac{5}{2}} \).
4. After differentiating, you will have an expression involving \( \frac{dy}{dx} \). Solve this expression for \( \frac{dy}{dx} \) to find the derivative, which represents the slope of the curve.
5. Finally, substitute the given point (4, 1) into the derivative expression to find the slope at that specific point.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx, which represents the slope of the curve at any point.

Slope of a Curve

The slope of a curve at a given point is defined as the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is represented by the derivative of the function at that point. For implicit functions, the slope can be found by evaluating dy/dx after applying implicit differentiation.

Evaluating Derivatives at a Point

Once the derivative (dy/dx) is found using implicit differentiation, the next step is to evaluate it at a specific point, in this case, (4, 1). This involves substituting the x and y values into the derivative expression to calculate the exact slope of the curve at that point, providing insight into the curve's behavior.