CEGEP CHAMPLAIN - ST. LAWRENCE
201-NYA-05: Differential Calculus
Patrice Camiré
Derivatives of Inverse Trigonometric Functions
1. Write down the differentiation formulas for the following inverse trigonometric functions.
(a) arcsin(x)
(c) arctan(x)
(e) arcsec(x)
(b) arccos(x)
(d) arccot(x)
(f) arccsc(x)
2. Differentiate the following functions.
(a) arcsin(x2 )
(b) arccos(3x)
2
(c) arcsin (x)
(d) arctan(2x + 1)
1
(e) arctan
x
(f) arcsec(4x)
(g) arcsec(ex )
(h) e
arcsin(x)
(i) x arccos(x)
(j)
√
(r) arctan( x)
(l) arcsin(ln(x))
(s) xarcsec(x)
√
(t) arcsec( x + 1)
√
(u) arcsin( x + 1)
arctan(x)
x
1
(k) arctan
x+1
(m) arcsin(x) arccos(x)
(v) arctan(cot(x))
(n) 2arcsec(3x)
(w) arctan(ex )
(o) log5 (arctan(2x))
p
(p) arcsin(x + 1)
(x) ln(arcsec(πx))
(q) arccos(e−6x )
(y) arcsin(xπ )
(z) arccos(π x )
3. Differentiate the following functions.
(a) y = arcsin(3x)
(b) y =
1
arccos(x2 )
2
(c) y = sin(arcsin(x))
1
arctan(x3 )
3
p
(e) y = 1 − x2 arcsin(x) − x
(d) y =
(i) y = x2 arctan(x)
√
(j) y = earcsin(
x)
(k) y = ln(arccos(ex ))
(l) y = x arcsec(x3 )
x+1
(m) y = arctan
x−1
(g) y = arccos(9x )
(n) y = arctan(sin(2x))
p
(o) y = arccos(x5 )
(h) y = 2arctan(x)
(p) y = 3arcsin(x)
(f) y = arctan(e4x )
1
(q) y = arcsin
x
1
(r) y = arctan
x2
(s) y = 3arctan(2x)
x)
(t) y = 2arctan(e
(u) y = arcsec(2x)
1
(v) y = arcsec(x4 )
4
(w) y = arctan(xex )
(x) y = arcsin(x ln(x))
4. Find
dy
.
dx
(a) y = xarctan(x)
(c) y = [sin(x)]arccos(x)
(b) y = [arcsin(x)]x
(d) y x = xarctan(y
2)
5. Differentiate the following functions. Simplify your answers as much as possible.
(a) sin(arctan(x2 ))
(e) sec(arctan(−x))
3
(b) tan(arccos(x ))
√
(c) cos(arcsin( x))
(g) tan(arcsin(x5 ))
(d) sec(arcsin(2x))
(h) cos(arctan(−3x))
(f) sin(arccos(1/x))
6. Provide the exact value of each trigonometric function at the given point.
(a) sin(0)
(d) sin(π/6)
(g) sin(π/4)
(j) sin(π/3)
(m) sin(π/2)
(b) cos(0)
(e) cos(π/6)
(h) cos(π/4)
(k) cos(π/3)
(n) cos(π/2)
(c) tan(0)
(f) tan(π/6)
(i) tan(π/4)
(l) tan(π/3)
7. Provide the exact value of each inverse trigonometric function at the given point.
(Use your answers to problem 6.)
√
(i) arctan(1)
(m) arcsin(1)
(a) arcsin(0)
(e) arccos( 3/2)
√
√
(j) arcsin( 3/2)
(n) arccos(0)
(b) arccos(1)
(f) arctan( 3/3)
√
(c) arctan(0)
(g) arcsin( 2/2)
(k) arccos(1/2)
√
√
(h) arccos( 2/2)
(d) arcsin(1/2)
(l) arctan( 3)
8. Find the equation of the tangent line to the function y = f (x) at the given value of x.
π
4
π
(b) y = cos(x) , x =
3
(a) y = sin(x) , x =
(c) y = tan(x) , x =
π
6
√
π2
(d) y = cos( x) , x =
4
9. Find the equation of the tangent line to the function y = f (x) at the given value of x.
√
1
2
2
(a) y = arcsin(x) , x =
(c) y = arccos(x ) , x =
2
2
√
(b) y = arctan(x) , x = 1
(d) y = arctan( x) , x = 3
Answers
d
1
arcsin(x) = √
dx
1 − x2
d
−1
(b)
arccos(x) = √
dx
1 − x2
1. (a)
2x
2. (a) √
1 − x4
−3
(b) √
1 − 9x2
(c)
(d)
(e)
(f)
(g)
2 arcsin(x)
√
1 − x2
1
2
2x + 2x + 1
−1
1 + x2
1
√
x 16x2 − 1
1
√
2x
e −1
earcsin(x)
(h) √
1 − x2
x
(i) arccos(x) − √
1 − x2
3
3. (a) √
1 − 9x2
−x
(b) √
1 − x4
(c) 1
(d)
(c)
d
1
arctan(x) = 2
dx
x +1
(d)
d
−1
arccot(x) = 2
dx
x +1
(j)
x2
+1
(l)
1
q
x 1 − ln2 (x)
(t)
1
2 x(x + 1)
(u)
1
p
2 −x(x + 1)
ln(2)2arcsec(3x)
√
(n)
x 9x2 − 1
2
(o)
2
ln(5)(4x + 1) arctan(2x)
(p)
x2
x2 + 1
−ex
√
arccos(ex ) 1 − e2x
(l) arcsec(x ) + √
3
x6 − 1
−1
+1
√
(w)
(x)
ex
1
+ e−x
1
√
x arcsec(πx) π 2 x2 − 1
πxπ−1
(y) √
1 − x2π
(q)
−1
√
|x| x2 − 1
(r)
−2x
x4 + 1
(s)
2 ln(3)3arctan(2x)
4x2 + 1
x)
ln(2)ex 2arctan(e
(t)
1 + e2x
(u)
1
x 4x2 − 1
2 cos(2x)
1 + sin2 (2x)
(v)
1
x x8 − 1
(o)
−5x4
p
2 1 − x10 arccos(x5 )
(w)
(x + 1)ex
1 + x2 e2x
(p)
3arcsin(x) ln(3)
√
1 − x2
1 + ln(x)
(x) p
1 − x2 [ln(x)]2
x2
√
−1
(v) −1
√
3
1
x2
−π x ln(π)
(z) √
1 − π 2x
earcsin( x)
(j) √
2 x − x2
(n)
2arctan(x) ln(2)
1 + x2
6e−6x
1 − e−12x
(i) 2x arctan(x) +
4e4x
1 + e8x
(h)
1
p
2 −x(x + 2) arcsin(x + 1)
(q) √
(m)
−9x ln(9)
(g) √
1 − 81x
1
2 x(x + 1)
(s) arcsec(x) + √
arccos(x) − arcsin(x)
√
(m)
1 − x2
1
x x2 − 1
−1
√
x x2 − 1
√
√
−1
(x + 1)2 + 1
−x arcsin(x)
√
(e)
1 − x2
(f)
(r)
(k)
(k)
x6
1
arctan(x)
−
+ 1)
x2
x(x2
d
arcsec(x) =
dx
d
(f)
arccsc(x) =
dx
(e)
√
√
dy
4. (a)
=
dx
dy
dx
(d)
dy
dx
5. (a)
ln(x)
arctan(x)
+
2
x +1
x
xarctan(x)
x
[arcsin(x)]x
ln(arcsin(x)) + √
1 − x2 arcsin(x)
ln(sin(x))
= cot(x) arccos(x) − √
[sin(x)]arccos(x)
1 − x2
2y ln(x) x −1
arctan(y 2 )
= ln(y) −
−
x
y4 + 1
y
dy
(b)
=
dx
(c)
x
1 + x2
1
q
(f)
x3 1 −
2x
(1 + x4 )3/2
(e) √
−3
√
x4 1 − x6
−1
(c) √
2 1−x
(b)
(d)
1
x2
5x4
(1 − x10 )3/2
−9x
(h)
(1 + 9x2 )3/2
(g)
4x
(1 − 4x2 )3/2
√
√
2/2
(j)
2/2
(n) 0
(i) 1
(k) 1/2
√
(l) 3
(d) π/6
(g) π/4
(j) π/3
(m) π/2
(b) 0
(e) π/6
(h) π/4
(k) π/3
(n) π/2
(c) 0
(f) π/6
(i) π/4
(l) π/3
6. (a) 0
(d) 1/2
√
(e) 3/2
√
(f) 3/3
(h)
7. (a) 0
(b) 1
(c) 0
(g)
√
3/2
√
2
2(4 − π)
8. (a) y =
x+
2
8
√
√
3
3 + 3π
(b) y = −
x+
2
6
√
4
3 3 − 2π
(c) y = x +
3
9
1
π
(d) y = − x +
π
4
√
√
π−2 3
2 3
9. (a) y =
x+
3
6
1
π−2
(b) y = x +
2
4
√
√
2 6
π+2 3
(c) y = −
x+
3
3
√
√
3
8π − 3 3
(d) y =
x+
24
24
√
(m) 1