4.3 derivatives of inv erse trig. functions
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Transcript of 4.3 derivatives of inv erse trig. functions
Derivatives of Inverse
Trigonometric Function
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
Find If
therefore
One more example
Find if
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.
Example:Confirm Differentiability of Inverse Function formula for the function
Solution:
and
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