In Exercises 51 and 52, find dp/dq.
q = (5p² + 2p)⁻³/²

Verified step by step guidance

1. Identify the function q in terms of p: \( q = (5p^2 + 2p)^{-\frac{3}{2}} \). We need to find \( \frac{dp}{dq} \), which is the reciprocal of \( \frac{dq}{dp} \).
2. Use the chain rule to differentiate q with respect to p. The chain rule states that if you have a composite function \( f(g(x)) \), then \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \).
3. Differentiate the outer function \( (u)^{-\frac{3}{2}} \) with respect to u, where \( u = 5p^2 + 2p \). The derivative is \( -\frac{3}{2}u^{-\frac{5}{2}} \).
4. Differentiate the inner function \( u = 5p^2 + 2p \) with respect to p. The derivative is \( 10p + 2 \).
5. Combine the derivatives using the chain rule: \( \frac{dq}{dp} = -\frac{3}{2}(5p^2 + 2p)^{-\frac{5}{2}} \cdot (10p + 2) \). Finally, find \( \frac{dp}{dq} \) by taking the reciprocal of \( \frac{dq}{dp} \).

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 need to differentiate the equation q = (5p² + 2p)⁻³/² with respect to p, treating q as a function of p. This method allows us to find the derivative dp/dq even when p is not isolated.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When differentiating q = (5p² + 2p)⁻³/², we apply the chain rule to handle the outer function (the exponent) and the inner function (the polynomial). This rule is essential for correctly calculating the derivative of complex expressions.

Reciprocal Relationships in Derivatives

In calculus, the relationship between the derivatives of two variables can be expressed as dp/dq = 1/(dq/dp). This reciprocal relationship is useful when we need to find dp/dq after calculating dq/dp. Understanding this concept helps in switching between the derivatives of dependent and independent variables effectively.