Common Calculus Mistakes
Derivative of inverse sine |
The Goal
Find
The Mistake
Find the mistake: (Roll the mouse over the math to see a hint in red) The Correction
(Roll the mouse over the area above to see the correction in blue) An Explanation 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 sin^{2}(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). The inverse sine function has the derivative shown in "The Correction". |
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