What is the derivative of #y=arccos(x )#?

Answer 1

The answer is:

#dy/dx = -1/(sqrt(1-x^2))#
This identity can be proven easily by applying #cos# to both sides of the original equation:
1.) #y = arccosx#
2.) #cos y = cos(arccosx)#
3.) #cos y = x#
We continue by using implicit differentiation, keeping in mind to use the chain rule on #cosy#:
4.) #-siny dy/dx = 1#
Solve for #dy/dx#:
5.) #dy/dx = -1/siny#
Now, substitution with our original equation yields #dy/dx# in terms of #x#:
6.) #dy/dx = -1/sin(arccosx)#
At first this might not look all that great, but it can be simplified if one recalls the identity #sin(arccosx) = cos(arcsinx) = sqrt(1 - x^2)#.
7.) #dy/dx = -1/sqrt(1 - x^2)#
This is a good definition to memorize, along with #d/dx[arcsin x]# and #d/dx[arctan x]#, since they appear quite frequently in advanced differentiation problems.
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Answer 2

The derivative of ( y = \arccos(x) ) is ( -\frac{1}{\sqrt{1-x^2}} ).

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Answer 3

The derivative of (y = \arccos(x)) is (-\frac{1}{\sqrt{1-x^2}}).

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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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