If y = K ^ ⁻¹ x 1+ K ^ ⁻¹ x and t = K ^ ⁻¹ x , then dy dt

Step-by-step solution

Given the function y = K^ ⁻¹ x 1 + K^ ⁻¹ x and the substitution t = K^ ⁻¹ x , we find dy dt . Substituting t into the expression for y yields y = t 1 + t . Differentiating with respect to t using the quotient rule gives: dy dt = (1)(1 + t) - t(1) (1 + t)^2 = 1 (1 + t)^2 . Replacing t with K^ ⁻¹ x provides the final derivative: dy dt = 1 (1 + K^ ⁻¹ x )^2 , which corresponds to option C .