Page 2 - Calculus

Calculus

Determining the derivative of a function from the first principles

Always use the following formula if asked to derive from the first principles:

′
( ) = lim ( +ℎ)− ( )
ℎ→0 ℎ

Examples

′
Determine ( )of the following functions from the first principles

2
i. ( ) = 2 − 3 + 1
3
ii. ( ) = 2x
3
iii. ( ) =
x 2

Recall the following
• Only use the method which involves ( + ℎ) when asked to derive from the
first principles.
• The following should be indicated on presentation of your answer f(x) and f(x+h)
• For ( + ℎ), replace with + ℎ in ( )
• On substituting 0 for ℎ, drop the lim
ℎ→∞

Solution
2
i. ( ) = 2 − 3 + 1
2
( + ℎ) = 2( + ℎ) − 3( + ℎ) + 1
2
2
= 2( + 2 ℎ + ℎ ) − 3 − 3ℎ + 1
2
2
= 2 + 4 ℎ + 2ℎ − 3 − 3ℎ + 1
′
= lim ( +ℎ)− ( )
ℎ→0 ℎ
2
2
2
2 +4 ℎ+2ℎ −3 −3ℎ+1− 2 +3 −1
= lim
ℎ→0 ℎ
2
4 ℎ+2ℎ −3ℎ
= lim
ℎ→0 ℎ
ℎ(4 +2ℎ−3)
=lim
ℎ→0 ℎ
= 4 + 2(0) − 3
∴ ( ) = 4 − 3
′

3
ii. ( ) = 2
( + ℎ) = 2( + ℎ)
3
= 2( + ℎ)( + ℎ)( + ℎ)
= 2(x+h)( + 2 ℎ + ℎ )
2
2