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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