Page 322 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

Notes
2
d y dy 1
x = sin(log )
x
dx 2 dx x
2
d y dy
x 2 x = y
dx 2 dx
(6)
Example: f(x) = e sin x f (x) is equal to:
x
Solution:
ax
f(x) = e sin bx
2
n
2 n/2
ax
-1
f (x) = (a + b ) .e sin(bx + n tan b/a)
a = 1, b = 1, n = 6
6
1
x
f 6 ( ) = (1 1) e x sin x 6tan (1)
3
x
e
= 8 sine x x 8 cosx
2
d n
Example: If In x n log x , then I n nI n 1 is equal to:
dx n
Solution:
d n
I = x n log x
n n
dx
1
y = x n log x y 1 x n nx n 1 log x
x
(y ) = nI + (n 1)!
1 n-1 n-1
I nI = (n 1)!
n n-1

Example: If y = ae + be + c, where a, b, c are parameters, then y”’ is equal to:
-x
x
Solution:
-x
x
y = ae + be + c
-x
y’ = ae be ;
x
x
y” = ae + be -x
x
y”’ = ae be -x
y”’ = y’

Example: If y = a cos (log x) + b sin (log x), where a, b are parameters, then x y” + xy’ is
2
equal to:
Solution:
y = a cos (log x) + b sin (log x)

xy’ = a sin(log x) + b cos(log x)

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