nth der
4
x
▶ Find the nth derivative of (x−1)(x−2)
h
i
16
1
▶ (−1)n n! (x−2)
n+1 − (x−1)n+1 , n > 2
nth der
▶ Find the nth derivative of xx+1
2
h
i−4
(−1)n n!
3
1
▶
− (x+2)n+1
4
(x−2)n+1
nth der
▶ Find the nth derivative of (x−1)4x
2 (x+1)
i
h
2(n+1)
1
1
▶ (−1)n n! (x−1)
n+1 − (x+1)n+1 + (x−1)n+2
nth der
1
▶ Find the nth derivative of (a)2 −(x)
2
h
i
1
1
1
▶ 2a
(−1)n n! (x+a)
n+1 − (x−a)n+1
nth der
▶ Find the tenth and the nth derivative of
x 2 + 4x + 1
x 3 + 2x 2 − x − 2
h
i
1
1
1
▶ 10! (x−1)
+
−
11
(x+1)11
(x+2)11
i
h
1
1
1
n
(−1) n! (x−1)n+1 − (x+1)n+1 − (x+2)n+1
nth der
x
▶ Find the nth derivative of 1+3x+2x
2
h
i
n
1
2
n
▶ (−1) n! (x+1)n+1 − (2x+1)n+1
nth der
1
▶ Find the nth derivative of x 4 −a
4
n n!
2 sin[(n+1) cot−1 (x/a)
1
1
▶ (−1)
− (x+a)n+1 −
2
2 (n+1)/2
4a3
(x−a)n+1
(a +x )
nth der
1
▶ Find the nth derivative of x 2 +x+1
.
n
n!
√
▶ 2(−1)
sin(n + 1)θ, where
3r n+1
p
√
r = (x 2 + x + 1), θ = cot−1 [(2x + 1)/ 3]
nth der
x
▶ Find the nth derivative of x 2 +x+1
▶ missing
nth der
3
▶ Prove that the value of the nth derivative of y = (x 2x−1) for
x = 0 is zero, if n is even, and −(n !) if n is odd and greater
that 1
▶ missing
nth der
▶ Show that the nth derivative of y = tan−1 x is
1
n−1
n 1
(−1) (n − 1)! sin n
π − y sin
π−y .
2
2
▶ missing
nth der
1+x
▶ Find the nth derivative of tan−1 , 1−x
▶ (−1)n−1 (n − 1)! sinn θ sin nθ where θ = cot−1 x.
nth der
x sin α
▶ Find the nth derivative of tan−1 1−x
cos α
▶ (−1)n−1 (n − 1)! cosecn α sinn θ sin nθ, where
θ = cot−1 [(x − cos α)/ sin x]
nth der
2x
▶ Find the nth derivative of tan−1 1−x
2
▶ 2(−1)n−1 (n − 1)! sinn θ sin nθ where θ = cot−1 x
nth der
2x
▶ Find the nth derivative of sin−1 1+x
2
▶ (−1)n−1 (n − 1)! sinn θ sin nθ where θ = cot−1 x.
nth der
▶ Find the nth derivative of tan−1
√
1+x 2 −1
x
▶ 12 (−1)n−1 (n − 1)! sinn θ sin nθ, where θ = cot−1 x
nth der
▶ If y = x(x + 1) log(x + 1)3 , prove that
3(−1)n−1 (n − 3)!(2x + n)
d ny
=
dx n
(x + 1)n−1
provided that n ≥ 3.
▶ missing
nth der
▶ Find the nth derivative of sin3 x.
▶ 43 sin x + 12 nπ − 34 n sin 3x + 21 nπ
nth der
▶ Find the nth derivative of cos x cos 2x cos 3x
▶
1
4
n
2 cos 2x + 12 nπ + 4n cos 4x + 21 nπ + 6n cos 6x + 12 nπ
nth der
▶ Find the nth derivative of sin 5x sin 3x
nπ
n
▶ 12 2n cos 2x + nπ
2 − 8 cos 8x + 2
nth der
▶ Find the nth derivative of sin2 x cos3 x.
▶
1
16
2 cos x + 21 nπ − 3n cos 3x + 21 nπ − 5n cos 5x + 12 nπ
nth der
▶ Find the nth derivative of e ax cos2 bx
h
ax
▶ e2 an + a2 + 4b 2 n/2 e ax cos 2bx + n tan−1
2b
2
nth der
▶ Find the nth derivative of e x sin4 x.
▶
1 x
8e
3 − 4n/2 cos 2x + n tan−1 2 − 17n/2 cos 4x + n tan−1 4
nth der
▶ Find the nth derivative of e x cos x cos 2x
x ▶ e2 (10)n/2 cos 3x + n tan−1 3 + (2)n/2 cos cos x + nπ
4
nth der
▶ Find the nth derivative of e 2x cos x sin2 2x.
▶
1 2n
4e
2 · 5n/2 cos x + n tan−1 /2 − 13n/2 cos 3x + n tan−1 3/2 − 2
leibnitz theorem
▶ Find the nth derivative of x n e x .
i
h
2
2
2
2
2
2
n−2 + . . . . n (n−1) ...1
▶ e x x n + n1! x n−1 + n (n−1)
x
2!
n!
leibnitz theorem
▶ Find the nth derivative of x 3 cos x.
▶ x 3 cos x + 21 nπ + 3nx 2 cos x + 12 (n − 1)π + 3n(n −
1)x cos x + 12 (n − 2)π
1
+n(n − 1)(n − 2) cos x + (n − 3)π
2
leibnitz theorem
▶ Find the nth derivative of e ax a2 x 2 − 2nax + n(n + 1) .
▶ an+2 · e ax · x 2
leibnitz theorem
▶ Find the nth derivative of e x log x.
▶
e x log x + n c1 x −1 − n c2 x −2 + n c3 2!x −3 − . . . . . . .. + (−1)n−1n cn (n −
leibnitz theorem
▶ If y = x 2 sin x, prove that
dny
nπ
nπ
2
2
dx n = x − n + n sin x + 2 − nx cos x + 2 .
▶ missing
leibnitz theorem
▶ If f (x) = x 2 tan x, prove that
f n (0) − n C2 f n−2 (0) + n C4 f n−4 (0) . . . = sin
▶ missing
nπ
.
2
leibnitz theorem
2
2
▶ Differentiate the equation: 1 − x 2 ddxy2 − x dy
dx + a y = 0.
▶ 1 − x 2 yn+2 − (2n + 1)xyn+1 + a2 − n2 yn = 0
leibnitz theorem
2
2
2
▶ Differentiate the equation: x 2 ddxy2 − x dy
dx + a − m y = 0. n
times with respect to x.
▶ x 2 yn+2 (2n + 1)xyn+1 + n2 + a2 − m2 yn = 0
leibnitz theorem
▶ If y = sin−1 x 2 , prove that
1 − x2
d 2y
dy
− 2 = 0.
−x
dx 2
dx
Differentiate the above equation n times with respect to x
and show that
1 − x 2 yn+2 − (2n + 1)xyn+1 − n2 yn = 0.
▶ missing
leibnitz theorem
▶ If u = tan−1 x, prove that
1 − x2
d 2u
du
− 2x
=0
dx 2
dx
and hence determine the values of the derivatives of u when
x = 0 (M.T .)
▶ un (0) = 0, if n is even ; and (−1)(n−1)/2 (n − 1) ! if n is odd.
leibnitz theorem
▶ If
y = sin m sin−1 x , show that
1 − x 2 yn+2 = (2n + 1)xyn+1 + n2 − m2 yn
and find yn (0).
▶ yn (0) =
0, if n is even
m 12 − m2 32 − m2 . . . (n − 2)2 − m2 , if n is odd
leibnitz theorem
im
h
p
▶ Find yn (0) when y = x + (1 + x 2 )
▶
y2n (0) = m2 m2 − 22
m2 − 42 . . . m2 − (2n − 2)2
2
y2n+1 (0) = m m2 − 12
m − 32 m2 − 52 . . . m2 − (2n − 1)2
leibnitz theorem
▶ If
−1 x
y = e m cos
show that 1 − x 2 yn+2 − (2n + 1)xyn+1 − n2 + m2 yn = 0
and find yn (0).
▶ missing
leibnitz theorem
−1
sin x
▶ If y = √
, show that
1−x 2
1 − x 2 yn+2 − (2n + 3)xyn+1 − (n + 1)2 yn = 0
▶ missing
leibnitz theorem
▶ If y = a cos(log x) + b sin(log x), show that
x 2 yn+2 + (2n + 1)xyn + 1 + n2 + 1 yn = 0
▶ missing
leibnitz theorem
▶ If y = sinh−1 x 2 , prove that
(1 + x 2 )yn+2 + 2xyn+1 + n2 yn = 0
▶ missing
leibnitz theorem
▶ If y = x 2 − 1 n , prove that
x 2 − 1 yn+2 + 2xyn+1 − n(n + 1)yn = 0 (
▶ Hint. log y = n log x 2 − 1 ⇒ ẏ1 x 2 − 1 = 2nxy
leibnitz theorem
▶ Prove that hthe nth dirrerential coofficient of xin (1 − x)n is
2
2
2
2
x
· (1−x)x2 +......
n!(1 − x)n 1 − n12 1−x
+ n 1(n−1)
2 ·22
▶ missing
leibnitz theorem
▶ If y = e tan
−1 x
prove that
1 + x 2 yn+1 + (2nx − 1)yn + n(n − 1)y(n−1) = 0
▶ missing
leibnitz theorem
−1
▶ If y = e a sin
x = prove that
1 − x 2 yn+2 − (2n + 1)xyn+1 − n2 + a2 yn = 0
▶ missing
leibnitz theorem
oi2
h
n
p
▶ If y = loge x + (1 + x 2 )
find (yn )0
▶ missing
leibnitz theorem
h
i
p
▶ If y = log x + (x 2 + a2 ) , prove that
a2 + x 2 y2 + xy1 = 0. Differentiate this differential equation
n2
n times and prove that limx→0 yn+2
yn = − a 2
▶ missing
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of sin x.
3
5
▶ x − x3! + x5! − . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of sec x.
2
4
6
▶ 1 + x2! + 5x4! + 61x
6! + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of sin−1 x.
3
5
7
▶ x + 12 · x3! + 32 · 12 x5! + 52 · 33 · 12 · x7! + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of e sin x .
2
4
▶ 1 + x + x2 − x8 + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of tan x.
3
5
x
▶ x + 23 + 15
+ ...
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of e x cos x .
2
3
4
5
x
▶ 1 + x + x2 − x3 − 11x
24 − 5 + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of
loge (1 + sin x).
2
3
4
5
x
x
▶ x − x2 + x6 − 12
+ 24
+ ...
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of
loge (1 + tan x).
▶ x − 12 x 2 + 23 x 3 − . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of
e x loge (1 + x).
2
3
5
▶ x + x2! + 2x3! + 9x5! + . . .
Maclaurin’s theorem
−1
▶ Apply Maclaurin’s Theorem to find the expansion of e a cos x .
o
n
3
2 4
2 2
▶ e aπ/2 1 − ax + a 2!x − a 1 + a2 x3! + 22 + a2 a 4!x + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of e x sec x
ex
or cos
x.
3
▶ 1 + x + x 2 + 2x3! + . . .
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to find the expansion of ax .
2
n
▶ 1 + x log a + x2! (log a)2 + . . . + xn! (log a)n
Maclaurin’s theorem
▶ Prove that
a2 − b 2 2 a a2 − 3b 2 3
e cos bx = 1 + ax +
x +
x
2!
3!
n/2
a2 + b 2
x n cos n tan−1 b/a + . . .
+ . . . .. +
n!
ax
▶ missing
Maclaurin’s theorem
▶ Prove that
e x cos x = 1 + x −
▶ missing
2x 3 22 x 4 22 x 5 23 x 7
−
−
+
+ ...
3!
4!
5!
7!
Maclaurin’s theorem
▶ Prove that
1
1
1
loge sec x = x 2 + x 4 + x 6 + . . .
2
12
45
▶ missing
Maclaurin’s theorem
▶ Apply Maclaurin’s Theorem to obtain the term upto x 4 in the
expansion of
loge 1 + sin2 x
4
▶ x 2 − 5x6 + . . .