Page 319 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 3. Find n derivative of following:
th
1 1 x 1 2x
(a) tan (b) sin
1 x 1 x 2

1 1 x 2
-1
(c) cos (d) tan x
1 x 2
4. If u = sin nx + cos nx, show that
1
r
u n r 1 ( 1) sin 2nx 2
r
where u denotes the r derivative of u with respect to x.
th
r
d n
5. If I (x n log )
x
n n
dx
Prove that I = n I + (n 1)!,
n n-1
Hence show that
1 1 1
I n n ! log x 1
2 3 n
Leibnitz’s theorem(only statement):

If y = u .v,
th
where u & v are functions of x possessing derivatives of n order then,
y = nC u v +nC u v + nC u v + ……… + nC u v + ……… + nC uv
n 0 n 1 n-1 1 2 n-2 2 r n-r r n n
! n
where, nCr
r !(n r )!
Properties:
1. nCr = nCn-r

2. nC = 1 = nCn
0
3. nC = n = nCn-1
1

Notes Generally we can take any function as u and any as v.( If y = u .v) But take v as the
function whose derivative becomes zero after some order.

Problems Based on Leibnitz’s theorem:
th
Obtain n derivatives of followings:
x n
2
x
3
1. x log x 2. 3. x e cos x
x 1

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