In Exercises 53 and 54, find dr/ds.


2rs - r - s + s² = -3

Verified step by step guidance

1. First, identify the equation given: 2rs - r - s + s² = -3. We need to find the derivative dr/ds.
2. To find dr/ds, we will use implicit differentiation. Differentiate both sides of the equation with respect to s.
3. When differentiating 2rs with respect to s, apply the product rule: d/ds(2rs) = 2r(ds/ds) + 2s(dr/ds).
4. Differentiate the remaining terms: d/ds(-r) = -dr/ds, d/ds(-s) = -1, and d/ds(s²) = 2s.
5. Set the derivative of the right side of the equation to zero, as d/ds(-3) = 0. Combine all differentiated terms to form an equation involving dr/ds, and solve for dr/ds.

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 one variable while treating the other variable as a function of the first. This allows us to find derivatives like dr/ds even when r and s are intertwined.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. In the context of finding dr/ds, it helps us relate the rates of change of r with respect to s, especially when r is expressed as a function of s.

Solving for Derivatives

Solving for derivatives involves isolating the desired derivative after applying differentiation techniques. In this problem, after differentiating the given equation implicitly, we will rearrange the resulting equation to express dr/ds explicitly. This step is crucial for obtaining the final answer and understanding the relationship between the variables.