Common Calculus Mistakes Chain Rule: Inverse Sine 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) 3. (Roll the mouse over the math to see a hint in red) 4. (Roll the mouse over the math to see a hint in red) 5. (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 two mistakes the notation sin-1(x) has been misunderstood to mean 1/sin(x). That interpretation might seem to be a natural extension of notation such as sin2(x), which does mean (sin(x))2, and sin-2(x), which does mean 1/(sin(x))2. However, for any function f(x), by convention when we write f-1(x), we mean instead the inverse function for f(x). So sin-1(x) means the inverse sine of x, that is, the function that undoes the sine function. It is not equal to 1/sin(x). Even under this false interpretation of sin-1(x), in the first mistake the power rule is used incorrectly and the derivative of x3 is not computed correctly. In the third mistake the wrong formula for the derivative of sin-1(x) was used, and the chain rule was used incorrectly, since the inner function x3 should have been substituted for x in the proposed derivative of sin-1(x). Learn the correct derivative formulas (visit Calculus Facts for help in learning the rules). The fourth mistake includes an extra factor that should not be there; so the chain rule was not used correctly. Finally, in the fourth mistake, the derivative of x3 is incorrect. Take care with even the "easy" parts of a problem. Home Page | Common Calculus Mistakes | Privacy Policy