# how to calculate this derivative?

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• rotchm
Lv 7
1 month ago

From highschool math, rewrite as (x+3)²/(3x-6).

Use now the quotient rule...

Or, from highschool math rewrite your original expression into partial fractions to get: y = 8/3 + x/3 + 25/(3x-6). And this is much easier to differentiate. [carefull, since its not quite the same function; their domains differ slightly. can you see why/where?]

• 1 month ago

y = (3x–6)⁻¹ / (x+3)⁻²

that is the same as

y = (x+3)² / (3x–6)

or

y =  (x²+6x+9) / (3x–6)

quotient rule

(f/g)' = (f'g–fg')/g²

f = x²+6x+9

f' = 2x+6

g = 3x–6

g' = 3

g² = (3x–6)²

y' = ((2x+6)(3x–6) – 3(x²+6x+9)) / (3x–6)²

y' = (6x² + 18x – 12x – 36 – 3x² – 18x – 27)) / (3x–6)²

y' = (3x² – 12x – 63) / (3x–6)²

y' = 3(x² – 4x – 21) / 3²(x – 2)²

y' = (x² – 4x – 21) / 3(x – 2)²

y' = (x – 7)(x + 3) / 3(x – 2)²