Easy -1.2 This is a straightforward application of the quotient rule to find dy/dx, followed by a simple sign analysis. The algebra is minimal, and showing the derivative is always negative requires only observing that the numerator is negative while the denominator is positive squared—a routine exercise with no problem-solving insight needed.
1 The equation of a curve is \(y = \frac { 1 + x } { 1 + 2 x }\) for \(x > - \frac { 1 } { 2 }\). Show that the gradient of the curve is always negative.
1 The equation of a curve is $y = \frac { 1 + x } { 1 + 2 x }$ for $x > - \frac { 1 } { 2 }$. Show that the gradient of the curve is always negative.
\hfill \mbox{\textit{CAIE P3 2013 Q1 [3]}}