Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 6: The Trigonometric Functions And Their Inverses – Section 6.2.2: Differentiation Of The Inverse
Trigonometric Functions

 

6.2.2
Differentiation Of The Inverse Trigonometric
Functions

 

 

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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:

 

 

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Problems & Solutions

 

1. Find the derivative of each of the following functions, simplifying the answer when appropriate.

 

  

 

Solution

 

 

 

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2. Find an equation of the line tangent to the curve y = arcsin(x/2) at the point x = –1.

 

Solution

 

 

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3.  Let f(x) = sin arcsin x.

 

     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].

 

 

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4. Let f(x) = arcsin sin x.

 

     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.

 

  

 

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5.  A careless mathematics professor asked his calculus class on their final examination to find dy/dx if y = arccos (1 +
     x2). What's wrong with this problem?

 

Solution

 

 

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