Differentiation SS2 Further Mathematics Lesson Note

LIMITS OF A FUNCTION

Notation For Limiting Value

Definition: lim𝑓(𝑥) = 𝐿

                 𝑥→𝑎

Intuitively, this means that, as x gets closer to a, f(x) gets closer and closer to L.

Right Hand Limit

lim𝑓(𝑥) = 𝐿

𝑥→𝑎

Intuitively, this means that, as x gets closer to a from right, f(x) gets closer and closer to L

Left Hand Limit

lim−𝑓(𝑥) = 𝐿

𝑥→𝑎

Intuitively, this means that, as x gets closer to a from left, f(x) gets closer and closer to L

Example: Find lim  x² – 25

                   𝑥→5      x – 5

Solution 

The numerator and denominator both approach 0. However, 

𝑥² − 25 = (𝑥 + 5)(𝑥 − 5)

Hence, 𝑥² − 25      = 𝑥 + 5

               𝑥 − 5

Thus,lim   𝑥² − 25

         𝑥→5  𝑥 − 5

= (𝑥 + 5)(𝑥 − 5)

          𝑥 − 5

= lim𝑥 + 5 = 5 + 5 = 10

   𝑥→5

DIFFERENTIATION FROM FIRST PRINCIPLE

Consider the function

𝑦 = 𝑓(𝑥) … (1)

Let ∆𝑦 be a small increase in y due to a small increase in ∆𝑥 in x

𝑦 + ∆𝑦 = 𝑓(𝑥 + ∆𝑥) … (2)

Subtracting (1) from (2)

𝑦 + ∆𝑦 − 𝑦 = 𝑓(𝑥 + ∆𝑥) − 𝑓 (𝑥)

∆𝑦 = 𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)

Divided both sides by ∆𝑥

∆𝑦

∆𝑥

=

𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)∆𝑥

Take the limits as ∆𝑥 → 0,

lim ∆y/∆x → dy/dx

NOTE: To differentiate by first principle means to work strictly by ordinary definition or the derived 

definition and not employing any other theorem.

GENERAL FORMULA METHOD 

For any function 𝑦 = 𝑎𝑥

𝑛 where a and n are constants

Then, differentiating y with respect to x is given as:

𝑑𝑦    = 𝑎𝑛𝑥ⁿ–¹

𝑑𝑥    

Pronounced as Dee y and – Dee x

Example 1: Differentiate 𝑦 = 𝑥⁶ with respect to x

Solution 

𝑦 = 𝑥⁶

𝑑𝑦    = 6𝑥⁵

𝑑𝑥

Example 2: Differentiate 

𝑦 = 4𝑥⁴ − 3𝑥³ − 2𝑥² + 𝑥 − 22 with respect to x.

Solution 

If y = 4𝑥⁴ − 3𝑥³ − 2𝑥² + 𝑥 − 22

dy/ dx = 4𝑥⁴–¹ – 3x³-¹ – 2x²-¹ + 1 × 1x¹-¹ – 22 × 0x⁰-¹

𝑑𝑦/ 𝑑𝑥 = 16𝑥³ − 9𝑥² − 4𝑥 + 1