Inverse Trigonometric Derivatives

Definition of the Inverse Trig Functions

Recall that we write

arcsin

to mean the inverse sin of x restricted to have values between -p/2 and p/2 (Note that sin x does not pass the horizontal line test, hence we need to restrict the domain.)  We define the other five inverse trigonometric functions similarly.

Trig of Arctrig Functions

Example:

Find tan(acrsin(x))

The triangle above demonstrates that

sin(t) = x/1 = opp/hyp.

Hence

Since the tangent is

We have

Exercise

Simplify

cos(arctan(2x))

Derivatives of the Arctrigonometric Functions

Recall that if f and g are inverses, then

What is

d
arctan(x)
dx

We use the formula:

Since

opp                x
tan(q) =                 =

we have

so that

 Theorem

Recall that

cos x  =  sin(p/2 - x)

hence

arccos x  =  p/2 - arcsin x

so

d/dx(arccos x) = d/dx[p/2 - arcsin x]

= -d/dx[arcsinx] =

Similarly:

d/dx(arccot(x)) = -1/(1 + x2)

d/dx(arccscx)

Example

Find the derivative of

cos(arcsinx)

Solution:

let

y = cosu,      u = arcsinx

y' = -sin u

= -sin(arcsin x) = x

We arrive at