Practice Exercise Sheet 1 - Trinity College Dublin
Self Assessment Questions
Differentiation-Solutions
Q1. Differentiate the following functions:
(i) [pic]
[pic]
(ii) [pic]
[pic]
(iii)[pic]
[pic]
(iv) [pic]
[pic]
[pic]
(v) [pic]
[pic]
[pic]
(vi) [pic]
[pic]
(vii) [pic]
[pic]
[pic]
NOTE: For parts (viii) and (ix) use the quotient rule:
If [pic], then [pic]
(viii) [pic]
Let [pic] and [pic]
[pic]
(ix) [pic]
Let [pic] and [pic]
[pic]
Q2. Differentiate the following functions:
(i) Y= X + 3
[pic]= 1
(ii) Y = 2X2 +7X +3
[pic]= 4X + 7
(iii) Y = X5
[pic] = 5X4
(iv) Y = X 1/n
[pic]
(v) [pic] can be written as [pic]
[pic]
(vi) Y = (X 2 +3) (X 3 –1 ) + 6X 2
use product rule…… and sum-difference rule
[pic] = (X 2 +3) (3X2) + (X 3 –1 ) (2X) + 12X
= 3X 4 + 9X 2 + 2X 4 – 2X + 12X
= 5X 4 + 9X 2 + 10X
(vii) Y = ((X +1) (X 3 + 3X)
can be re-written as Y = (X ½ +1) (X 3 + 3X)
applying the product rule….
[pic] = (X ½ +1)( 3X 2 + 3) + (X 3 + 3X) (½ X –½ )
multiplying out…remember, xa.xb = x a+b so e.g. X ½. 3X 2 = 3X 5/2
= (3X 5/2 + 3 X ½ + 3X 2 + 3) + (½ X 5/ 2 + 3/2X ½ )
= 7/2 X 5/ 2 + 9/2 X ½ + 3X 2 + 3
(viii) [pic]
applying the quotient rule….. If [pic], then [pic]
Let u = X 2 +1 and v = X 2 – 2X + 1
[pic]
(ix) Y = 1/X (X 4 – 2X –1)
[pic] = 1/X (4X 3 – 2 ) + (X 4 – 2X –1) (- 1 / X 2 )
= 4X2 – 2/ X - X 2 + 2X –1 + (1/ X 2 )
= 4X 2 – 2X –1 – X 2 + 2X –1 + X - 2
= 3X 2 + X - 2
Q3. Differentiate the functions:
NOTE: To differentiate these functions, use the chain rule:
If [pic] is a function of [pic] and [pic] is a function of [pic]then [pic]
(i) [pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
(ii) [pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
(iii) [pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
(iv) [pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
(v) [pic]
[pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
(vi) [pic]
Let [pic]
[pic]
[pic]
[pic]
[pic]
Differentiation of Logs and Exponentials
Q4. Differentiate the following functions:
NOTE: To differentiate exponentials use the following rule:
If [pic] then [pic]
(i) [pic]
[pic]
(ii) [pic]
[pic]
(iii) [pic]
Use the Chain Rule
Let [pic]
[pic]
[pic]
[pic]
[pic]
(iv) [pic]
Simplify using rules of indices
[pic]
[pic]
(v) [pic]
Use rule of logs to simplify
[pic]
[pic]
(vi) [pic]
Use Chain Rule
Let [pic]
[pic]
[pic]
[pic]
[pic]
Differentiate the following functions:
Use rule y = ln x ( [pic]
(vii) [pic]
Use the product rule:
If [pic], then [pic]
Let [pic] and [pic]
[pic]
[pic]
(viii) [pic]
Use Chain Rule
Let [pic]
[pic]
[pic]
[pic]
[pic]
(ix) [pic]
Use Chain Rule
Let [pic]
[pic]
[pic]
[pic]
[pic]
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