Some problems provide the opportunity for more than one mistake.

The Goal

Find

The Mistakes

Find the mistakes:

1.

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2.

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3.

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4.

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5.

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A Correct Solution

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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.