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1)
Derivative the basic rule:
Where the Function is 𝑦 = 𝑥 n
The Derivative 𝑑𝑦 /𝑑𝑥 = n x 𝑥 (n-1)

Example:Function
𝑦 =𝑥 2

Derivatives
𝑑𝑦 / 𝑑𝑥 = 2 x 𝑥 (2-1) = 2𝑥

Questions and Answers
I
...


𝑦 =𝑥 -2

𝑑𝑦 / 𝑑𝑥 = -2 x 𝑥 (-2-1) = -2 𝑥 -3

III
...


𝑦 =8𝑥 4

𝑑𝑦 / 𝑑𝑥 =4 x 8𝑥 (4-1) = 32𝑥 3

II
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𝑦 =1/2𝑥 9

𝑑𝑦 / 𝑑𝑥 =9 x 1/2𝑥 (9-1) = 4
...


𝑦 = 6𝑥 4+27

𝑑𝑦 / 𝑑𝑥 = 4 x 6𝑥 (4-1) = 24𝑥 3

II
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𝑦= 1/6𝑥 2 - 12𝑥 5

𝑑𝑦 / 𝑑𝑥 = 2 x 1/6𝑥 (2-1) - 5 x 12𝑥 (5-1)
= 1/3𝑥 - 60𝑥 4

II
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𝑑𝑛/ 𝑑𝑥
𝑛2
= 𝑥 2(21𝑥 2+1) – (7𝑥 3+𝑥) 2𝑥
(𝑥 2)2
= 21𝑥 4 + 𝑥2 -14𝑥 4 - 2𝑥 2
𝑥4
= 7𝑥 4 - 𝑥 2
𝑥4

𝑉𝐼𝐼) 𝑦 = 4𝑥 2 + 3

𝒎

2𝑥 -1

𝒏

𝑑𝑦 / 𝑑𝑥 = 𝑛 x 𝑑𝑚/ 𝑑𝑥 – 𝑚 x 𝑑𝑛/ 𝑑𝑥
(2𝑥 -1)2
= (2𝑥-1)8𝑥 – (4𝑥 2+3)2
4𝑥 2 - 4𝑥 +1
= 16𝑥 2 - 8𝑥 - 8𝑥 2 - 6
4𝑥 2 - 4𝑥+1
= 8𝑥 2-8𝑥-6
4𝑥 2-4𝑥+1

𝑉𝐼𝐼𝐼) 𝑦 = 4 - 5𝑥
(2𝑥-1) (3𝑥+2)

𝒎
𝒏

𝑑𝑦 / 𝑑𝑥 = n x 𝑑𝑚/ 𝑑𝑥 – m x 𝑑𝑛/ 𝑑𝑥
𝑛2
= [(2𝑥-1) (3𝑥+2)]-5 – [(4-5𝑥) (12𝑥+1)]
[6𝑥 2 + 𝑥 -2]2
= (6𝑥 2+4𝑥-3𝑥-2)-5 - (48𝑥 + 4 - 60𝑥 2 - 5𝑥)
36𝑥 4 + 12𝑥3 - 23𝑥 2 + 4

= (-30 𝑥2 - 20𝑥 + 15𝑥+10) (-48𝑥 – 4 + 60𝑥 2 + 5𝑥)

36𝑥 4 + 12𝑥3 - 23𝑥 2 + 4
= -30𝑥 2 - 5𝑥 +10 - 43𝑥 + 60𝑥 2 - 4
36𝑥 4 + 12𝑥3 - 23𝑥 2 + 4
= 30𝑥 2 - 48𝑥 + 6
36𝑥 4 + 12𝑥3 - 23𝑥 2 + 4
7)
Derivation function of a function where 𝒚= (2𝒙+6)3 and the expression in the
brackets is a derivatization functions say that is 𝒎 = (2𝒙+6) the code
expression can be written as 𝒚=𝒎3
In that cases he rule for derivation is 𝒅𝒚 / 𝒅𝒙 = 𝒅𝒚 / 𝒅𝒎 x 𝒅𝒎/ 𝒅𝒙
Which is known as the chain rule
...
𝑑𝑚/ 𝑑𝑥
= 4𝑥 x 1/𝑥 + 𝑥 x 4
= 4𝑥 (1/𝑥+1)
g)
𝒎
𝒏
𝑦 = (1+ 𝑥2) 𝑒ax
𝑑𝑦 / 𝑑𝑥 = (1+𝑥 2) 𝑎𝑒ax + 𝑒ax x 2x
𝑒ax (1+𝑥 2)x a +2𝑥

h)
𝒎 𝒏
𝑦 = e𝑥 In 𝑥
𝑑𝑦 / 𝑑𝑥 = e𝑥 x 1/𝑥 + 𝑥 x 𝑒x
𝑒x (1/𝑥+𝑥)



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