How do you find the derivative of Inverse trig function #y=arcsec(1/x) #?

Answer 1

#d/dx(arcsec(1/x))=-1/sqrt{1-x^2}# for #-1 < x < 1# with #x!=0#.

First, let's find the derivative of #arcsec(x)# by applying the Chain Rule to the equation #sec(arcsec(x))=x# (and using the fact that #d/dx(sec(x))=sec(x)tan(x)#)
#d/dx(sec(arcsec(x)))=d/dx(x)=1\Rightarrow sec(arcsec(x))tan(arcsec(x))d/dx(arcsec(x))=1#
#\Rightarrow d/dx(arcsec(x))=1/(x*sqrt{x^2-1})# (draw a right triangle with one non-right angle labeled #arcsec(x)# and use the Pythagorean Theorem to help you do this last simplification).

Now the Chain Rule implies:

#d/dx(arcsec(1/x))=d/dx(arcsec(x^{-1}))=(-1x^{-2})/(x^{-1}*sqrt{(x^{-1})^2-1})#
This simplifies, after multiplying the top and bottom of the last fraction by #x^2#, to:
#d/dx(arcsec(1/x))=-1/sqrt{1-x^2}#
Evidently this implies that #arcsec(1/x)=arccos(x)+C# for all allowed #x# and some #C#. In fact, it can be shown that #C=0# and #arcsec(1/x)=arccos(x)# for all #-1\leq x\leq 1# and #x!=0#.

Try graphing them both to check this!

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Answer 2

To find the derivative of the inverse trigonometric function ( y = \text{arcsec}(1/x) ), you can use the chain rule, which states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

So, for ( y = \text{arcsec}(1/x) ), the derivative is:

[ \frac{dy}{dx} = -\frac{1}{x \cdot |x| \cdot \sqrt{x^2 - 1}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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