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**Derivative of a function**

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**lim x-6 x2-36 lim x2-12x+36 x2-7x+6 lim x2-81 9-x x ⟶ 6 x ⟶ 9 x ⟶ 1**

Warm-up lim x-6 x ⟶ 6 x2-36 1 12 ∴ limit exists lim x2-12x+36 x ⟶ 1 x2-7x+6 = -5/0 , ∴ limit DOES NOT exist lim x2-81 x ⟶ 9 9-x =-18 , ∴ limit exists

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**Definition of the Derivative**

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**EXAMPLE: Find the derivative of f(x) = 3x2 - 2x**

f’(x) = [3(x+h)2 - 2(x+h)] - [3x2 - 2x] h f’(x) = [(3x2 +6xh + 3h2 ) - (2x+2h)] - [3x2 - 2x] h f’(x) = 3x2 +6xh + 3h2 - 2x - 2h - 3x2 + 2x h f’(x) = 6xh + 3h2 - 2h h f’(x) = (6x + 3h - 2) f’(x) = h (6x + 3h - 2) h f’(x) = (6x - 2)

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**PRACTICE 1. f(x)= 5x- 8 2. f(x)=-3x + 9 3. f(x)= 12x + 3x - 10 5 -3 15**

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