Final answer:
To find the derivative (dy)/(dx) for the function y = ((1 + 2x)/(1 + x))^2, apply the chain rule and the quotient rule to obtain the result via substitution and simplification.
Step-by-step explanation:
The student has asked to find the derivative (dy)/(dx) given the function y = ((1 + 2x)/(1 + x))^2. To solve this, we can use the chain rule and the quotient rule for derivatives. The chain rule is applied when we have a function raised to a power, and the quotient rule is used when we have a division of functions.
Firstly, let's set u = (1 + 2x)/(1 + x). The derivative of y with respect to x when y = u^2 is given by dy/dx = 2u(du/dx). Next, we calculate du/dx using the quotient rule which is du/dx = ((1+x)(2) - (1 + 2x)(1))/(1+x)^2. Simplify this expression to find du/dx and substitute it back into our equation for dy/dx. Through substitution and simplification, we can find the value of (dy)/(dx).