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1.
Differentiation 
The Derivatives Of Arcsine And
Arccosine
Consequently:

Similarly:

Remark that (d/dx) arccos x is the negative of (d/dx) arcsin x.
The Derivatives Of Arctangent
And Arccotangent

Note that Eq. [1.3] is valid for all x in R. This is consistent with the fact that dom(arctan x) is R.
Similarly:

Remark that (d/dx) arccot x is the negative of (d/dx) arctan x.
The Derivatives Of Arcsecant And
Arccosecant
So, since sec y = x, we obtain:

Similarly:

Remark that (d/dx) arccsc x is the negative of (d/dx) arcsec x.
Example 1.1
Differentiate each of the following functions, simplifying the answer when appropriate.
Solution
EOS
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2. Relationships Between Inverse Trigonometric Functions 
for all x
in (–1, 1). We saw in Section
4.1 Theorem 6.1 that if a function f is continuous on [a, b] and its
derivative is 0 on
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3. Avoiding Confusions In Memorizing The Derivatives 
We observe that:
The derivative of 
is simply the negative of the derivative of 
arccos x [Eq. [1.2]] 
arcsin x [Eq. [1.1]], 
arccot x [Eq. [1.4]] 
arctan x [Eq. [1.3]], 
arccsc x [Eq. [1.6]] 
arcsec x [Eq. [1.5]]. 
As for (d/dx) arctan x, there should be no confusion with any other derivative.
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4. Why Bother With All Of These “Inverse” Stuffs? 
The derivative of arcsin x is:
We've got three new antiderivatives: arcsin, arctan,
and arcsec. Well, that's why we bother with all of these “ inverse”
stuffs.
Note that we didn't “
boast ”
about the derivatives of the inverse cofunctions arccos x, arccot x, and arccsc x. The reason
is that they're simply the negatives of the derivatives of arcsin
x, arctan x, and arcsec x respectively, as observed in Part
3, and thus provide no new antiderivatives. For example, we know that:
Problems
& Solutions 
1. Find the derivative of each of the following functions, simplifying the answer when appropriate.
Solution
2. Find an equation of the line tangent to the curve y = arcsin(x/2) at the point x = –1.
Solution
a. Show that f '(x) = 1 by
calculating it directly from the given expression.
b. Simplify f(x). Find f '(x) using this
simplified form of f(x).
c. Sketch a graph of f(x).
Note: Also see Problem & Solution 4.
Solution
c. The domain of f is [–1, 1]; its range is also [–1, 1].
a. Show that:
where k
is any integer, by calculating f '(x) directly
from the given expression.
b. Simplify f(x). Find f '(x) using this
simplified form of f(x).
c. Sketch a graph of f(x).
Note. Also see Problem & Solution 3.
Solution
a. We have:
It follows that:
c.
5. A careless mathematics
professor asked his calculus class on their final examination to find dy/dx if y = arccos (1 +
x^{2}). What's wrong with this problem?
Solution
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