What is the derivative of #f(x)=cos^-1(x^3)# ?

Answer 1
A side comment to start with: the notation #cos^-1# for the inverse cosine function (more explicitly, the inverse function of the restriction of cosine to #[0,pi]#) is widespread but misleading. Indeed, the standard convention for exponents when using trig functions (e.g., #cos^2 x:=(cos x)^2# suggests that #cos^(-1) x# is #(cos x)^(-1)=1/(cos x)#. Of course, it is not, but the notation is very misleading. The alternative (and commonly used) notation #arccos x# is much better.
Now for the derivative. This is a composite, so we will use the Chain Rule. We will need #(x^3)'=3x^2# and #(arccos x)'=-1/sqrt(1-x^2)# (see calculus of inverse trig functions ). Using the Chain Rule: #(arccos(x^3))'=-1/sqrt(1-(x^3)^2) \times (x^3)'=-(3x^2)/sqrt(1-x^6) #.
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Answer 2

To find the derivative of ( f(x) = \cos^{-1}(x^3) ), we use the chain rule. The derivative is:

[ f'(x) = -\frac{1}{\sqrt{1 - (x^3)^2}} \cdot 3x^2 \cdot (-\sin(\cos^{-1}(x^3))) ]

Simplified, it becomes:

[ f'(x) = \frac{3x^2}{\sqrt{1 - x^6}} \cdot \sin(\cos^{-1}(x^3)) ]

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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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