Download - 4.3 derivatives of inv erse trig. functions
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Derivatives of Inverse
Trigonometric Function
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Inverse Trig Functions
Some trig functions domains’ have to be restricted in order for them to have an inverse function – why?
Only functions that are 1-to-1 can have inverse functions
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Find If
therefore
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One more example
Find if
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Differentiability of Inverse Functions
If f(x) is differentiable on an interval I, one may wonder whether f-1(x) is also differentiable? The answer to this question hinges on f'(x) being equal to 0 or not . Indeed, if for any , then f-1(x) is also differentiable. Moreover we have
Using Leibniz's notation, the above formula becomes
which is easy to remember.
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Example:Confirm Differentiability of Inverse Function formula for the function
Solution:
and
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MONOTONIC FUNCTIONS:Suppose that the domain of a function f is on an open interval I on which f’(x) > 0 or on which f’(x) < 0. Then f is one-to-one, f-1(x) is differentiable at all values of x in the range of f.
Example:
Consider the function .Show that f(x) is one-to=one function.
Solution:
Since f’(x) > 0 on the entire domain, f(x) is monotonic, therefore it has an inverse