Help me with differentiation!!

👨🏻‍🏫 學術台數學

[匿名]

2012-11-20 10:16 am

What is the derivative of f(x)= x* abs(x)

Please show complete steps on how to deal with the absolute value symbol
Thanks!!

更新1:

係唔係 f(x) cannot be differentiated at x=0 可不可以用 first derivative rule計

回答 (3)

Pak Wai

2012-11-20 10:53 am

✔ 最佳答案

f(x) = x|x|
f'(x)
= x(d/dx |x|) + |x| ... (Product rule)
= x(x/|x|) + |x|
= (x^2)/|x| + |x|
= (|x|^2)/|x| + |x|
= |x| + |x|
= 2|x|

Note:
In y = |x|
since |x|^2 = x^2
-> |x| = (x^2)^(1/2)
dy/dx
= [(1/2)(x^2)^(-1/2)](2x)
= x/(x^2)^(1/2)
= x/|x|

i.e. d/dx |x| = x/|x|

參考: Myself

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Jian Hua

2012-11-21 12:04 am

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參考: Math Use Bi Teaching (86) 10

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50418129

2012-11-20 3:58 pm

case(i), x >= 0,
f(x) = x(x) = x^2 => f '(x) = 2x

case(ii), x < 0,
f(x) = x(-x) = -x^2 => f '(x) = -2x

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