Question

Researchers have found a correlation between respiratory rate and body mass in the first three years of life. This correlation can be expressed by the function log R(w) = 1.84 - 0.42 log (w), where w is the body weight (in kilograms) and R(w) is the respiratory rate (in breaths per minute). Find R'(w) using implicit differentiation. Find R'(w) by first solving the equation for R(w). Discuss the two procedures. Is there a situation when you would want to use one method over another? R'(w) = (Simplify your answer. Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.)

Answer 1

To find R'(w) using implicit differentiation, take the derivative of both sides of the equation and solve for R'(w). Alternatively, you can solve the equation for R(w) and then differentiate it to find R'(w).

Step-by-step explanation:

To find R'(w) using implicit differentiation, we can start by taking the derivative of both sides of the equation log R(w) = 1.84 - 0.42 log (w). The derivative of log R(w) with respect to w is 1/R(w) * R'(w), and the derivative of 1.84 - 0.42 log (w) with respect to w is -0.42/w. So, we have 1/R(w) * R'(w) = -0.42/w. Rearranging this equation, we can solve for R'(w) by multiplying both sides by R(w), resulting in R'(w) = -0.42 * R(w) / w.

To find R'(w) by first solving the equation for R(w), we can start with the equation log R(w) = 1.84 - 0.42 log (w). By taking the antilogarithm of both sides, we get R(w) = 10^(1.84 - 0.42 log (w)). Then, to find R'(w), we can differentiate R(w) with respect to w using the power rule and chain rule. The final result is R'(w) = -0.42 * R(w) / w.