Some problems provide the opportunity for more than one mistake.

The Goal

Find dy/dx, where x and y are related by the equation

The Mistakes

Find the mistakes:

1.

Need a hint? Look carefully at the red part:

2.

Need a hint? Look carefully at the red part:

A Correct Solution

Explanations

In the first mistaken solution the derivative of ln(xy) with respect to x was computed incorrectly. By the chain rule the derivative of ln(Q) is 1/Q times the derivative of Q; in this case, 1/(xy) times the derivative of xy with respect to x. The mistaken solution incorrectly has the derivative of ln(Q) as 1/Q'. In the next step there is a second mistake - the reciprocal of the left side is taken, but the right side is left unchanged.
In the second mistaken solution the student started by using a logarithm rule to simplify the ln(xy) term. This made it easier to find the derivative. Alas, in the step following the differentiation, 1/x was subtracted from the left side but the right side was multiplied by x.