6 Further appli ations, using hain, produ t and quotient rules 21
6.1 A tivity: PE 6.16 (MO p 46) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Elasti ity and the derivative 24
7.1 A tivity: PE 6.17 (MO p 46) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2
, DSC1520/MO001
1 The power rule
The power rule states that if we have the fun tion y = f (x) = xn , then the derivative of the fun tion
is
dy
= f ′ (x) = nxn−1 .
dx
The power rule in words: Multiply by the power of the variable and subtra t one from the power.
1.1 Examples
The derivative of ea h of the following fun tions is found by using the power rule:
Function (simplify) Derivative
dy
y = x3 dx = 3 × x3−1 = 3x2
f (x) = 5x4 f ′ (x) = 5 × 4x4−1 = 20x3
g(x) = x3 + 5x4 g ′ (x) = 3x2 + 20x3
1
f (x) = x = x−1 f ′ (x) = −1 × x−1−1 = −x−2 = − x12
4 dy
y= x3
= 4x−3 dx = 4 × −3x−3−1 = −12x−4 = − x124
1 4
G′ (x) = − x12 − − x124 = − x12 + 12
G(x) = x − x3 x4
√ 1 1 2
f (x) = x = x3
3
f ′ (x) = 31 x 3 −1 = 13 x− 3 = 1
2 = √1
3 2
3x 3 3 x
√ 1 1 3
f (x) = 8 4 x = 8x 4 f ′ (x) = 8 × 41 x 4 −1 = 2x− 4 = √2
4 3
x
√ √ 1 2
g(x) = 3
x+84x g ′ (x) = √
3 2 + √
4 3
3 x x
1.2 A tivity: PE 6.1 (MO p 41)
1. Q3( )
1
Given the fun tion y = 10 + 5x + x2
. After simpli
ation,
y = 10 + 5x + x−2 .
Now,
dy
= 0 + 5x1−1 + (−2)x−2−1
dx
= 5x0 − 2x−3
2
= 5 − 3.
x
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