Section 5.6
Inverse Trigonometric
Functions: Differentiation
Section 5.6
Inverse Trigonometric Functions: Differentiation

We have been examining functions known
as transcendental functions. These are
functions that are not ‘algebraic’ in the
sense that they can be expressed strictly in
terms of x and y. The most common
transcendental functions are the
trigonometric functions whose Calculus
properties we have already examined.
Section 5.6
Inverse Trigonometric Functions: Differentiation



What we have not examined are the
properties of their inverse functions.
You should be familiar with the notation of
the inverse trig functions from your
precalculus days.
The notation y = arcsinx means two things
to us:


y is the inverse function of sinx
y is an angle and x is a ratio
Section 5.6
Inverse Trigonometric Functions:
Differentiation

So, the challenge to us right now is to find a
derivative rule for these inverse functions.
Similar to our approach with exponential
functions, we will work with the inverse
function.
d
dy
y  arcsin x  sin y  x  sin y  x   cos y
1
dx
dx
dy
1

dx cos y
Section 5.6
Inverse Trigonometric Functions:
Differentiation

It feels vaguely unsatisfying to have the
derivative defined in terms of y rather than
x. Using our Pythagorean identity we can
translate cosy and rewrite the derivative as
follows:
dy
1
1
1



dx cos y
1  sin 2 y
1  x2
Section 5.6
Inverse Trigonometric Functions:
Differentiation

We can go through similar processes to find the
derivatives of the other inverse trig functions. I will
summarize them here and we will work through
some of these examples in the coming days:
du
dy
dx
y  arccos u 

dx
1 u2
du
dy
dx
y  arc sec u 

dx u u 2  1
du
dy
dx
y  arcsin u 

dx
1 u2
du
dy
dx
y  arc csc u 

dx u u 2  1
du
dy
y  arctan u 
 dx2
dx 1  u
du
dy
dx
y  arc cot u 

dx 1  u 2
Section 5.6
Inverse Trigonometric Functions:
Differentiation

Find the derivatives of the following
functions. Feel free to confer with your
neighbor while I sit quietly for a few
minutes:
dy
y  arcsin x  arccos x 

dx
dy
y  arctan x 

dx
x
dy
y  arctan x 


2
1 x
dx
Section 5.6
Inverse Trigonometric Functions:
Differentiation

Finally, we can rewrite some of the transcendental
functions as algebraic expressions. We did so when
finding the derivatives. Rewrite the following so that
the trigonometric or inverse trigonometric function is
removed.
sec  arctan 4 x 
cos arcsin  x  1 