A curve is defined by the parametric equations x = 3^(-t) + 1, y = 2 x 2^(t). Show that dy\dx = -2 x 3^(2t).

Write 3^(t) as an expression involving x : We can rewrite x = 3^(-t) + 1 as x - 1 = 3^(-t) ; equivalently, 3^(t) = (x-1)^(-1). Substitute this expression into y, to write y in terms of x: y = 2 x 3^(t) = 2 x (x-1)^(-1). Differentiate y with respect to x, using the power rule:dy\dx = -2(x-1)^(-2). Substitute in the expression for 3^(t):dy\dx = -2(x-1)^(-2) = -2 x (3^(t))^(2) = -2 x 3^(2t)