Common Calculus Mistakes Implicit Differentiation: Multiple Mistakes 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. (Roll the mouse over the math to see a hint in red) 2. (Roll the mouse over the math to see a hint in red) A Correct Solution (Roll the mouse over the area above to see the corrections in blue) 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. Home Page | Common Calculus Mistakes | Privacy Policy