Some problems provide the opportunity for more than one mistake.

The Goal

Find

where w and L are constants.

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)

3.

(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

The presence of parameters, or arbitrary constants, like w and L, seem to give students all kinds of trouble in calculus (or any prior class). The correct approach is to treat these constants as numbers whose values happen not to be specified. The constant multiple rule should then be used to pull these constants out in front of any expression that is to be differentiated.

The first mistake is just one of forgetting to multiply the 2 arising from differentiating x² by the constant 1/2.

In the second mistake the student has attempted (incorrectly) to apply the product rule to differentiate each of the two terms; the constant L has been wrongly treated as a variable (in particular, as being equivalent to x). Moreover, even taking L as a variable in this way, the product rule was used incorrectly on the first term.

In the third mistake it appears that the student misinterpreted the meaning of the fraction notation by assuming that all of the parameters and x factors appear in the respective denominators (although the "2" in the second term was for some reason placed in the numerator).

The correct solution shows how the constant multiple rule boils the problem down to finding the derivatives of x and x².