Some problems provide the opportunity for more than one mistake.

The Goal

Find

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 both cases the formula for the derivative is wrong. The first mistake mimics the pattern for the trigonometric functions - but the derivatives of the hyperbolic functions don't agree with the corresponding trigonometric derivatives in sign in all cases. In the second mistake the student has confused the definition of hyperbolic cosine (in terms of exponentials) with the derivative of hyperbolic cosine.

Basic derivative formulas should be learned. For hyperbolic cosine there are two choices: memorize the correct derivative (try the flashcards at Calculus Facts); or derive the derivative using the definition of hyperbolic cosine whenever this derivative is needed: