Standard +0.8 This implicit differentiation problem requires differentiating ln(xy) using the chain rule, then solving for dy/dx algebraically. Finding the second derivative requires the quotient rule and careful substitution. While systematic, it demands more algebraic manipulation and care than standard implicit differentiation questions, placing it moderately above average difficulty.
5 The positive variables \(x\) and \(y\) are related by
$$y = x ^ { 2 } + 2 \ln ( x y )$$
Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when both \(x\) and \(y\) are equal to 1 .
5 The positive variables $x$ and $y$ are related by
$$y = x ^ { 2 } + 2 \ln ( x y )$$
Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when both $x$ and $y$ are equal to 1 .
\hfill \mbox{\textit{CAIE FP1 2007 Q5 [7]}}