Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. d/dx(tan^−1 x) =sec² x

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To determine whether the statement $ \frac{d}{dx}(\tan^{-1} x) = \sec^2 x $ is true, we need to find the derivative of $ \tan^{-1} x $.

Recall that $ \tan^{-1} x $ is the inverse function of $ \tan x $. The derivative of $ \tan^{-1} x $ is given by the formula $ \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2} $.

Compare the derivative $ \frac{1}{1 + x^2} $ with $ \sec^2 x $. Note that $ \sec^2 x = 1 + \tan^2 x $, which is different from $ \frac{1}{1 + x^2} $.

Since $ \frac{1}{1 + x^2} $ is not equal to $ \sec^2 x $, the statement $ \frac{d}{dx}(\tan^{-1} x) = \sec^2 x $ is false.

Therefore, the correct derivative of $ \tan^{-1} x $ is $ \frac{1}{1 + x^2} $, not $ \sec^2 x $. This serves as a counterexample to the given statement.

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