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Is the function f (x) = |x2 - x| differentiable at x = 2. If yes find it’s derivative.

Is the function f (x) = |x2 - x| differentiable at x = 2. If yes find it’s derivative.

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
Lets 1stcheck the differentiability at x = 2
Left Hand Derivative (LHD) = Right Hand Derivative (RHD)
f(x) = |x^{2}-x|
LHD = \lim_{h\rightarrow 0}\frac{f(2-h)-f(2)}{-h}
f(x) = x^{2}-x, x < 0
= x-x^{2}, 0 < x < 1
= x^{2}-x, x > 1
LHD = \lim_{h\rightarrow 0}\frac{[(2-h)^{2}-(2-h)]-f(2)}{-h}
LHD = \lim_{h\rightarrow 0}\frac{h^{2}-3h}{-h} = 3
RHD = \lim_{h\rightarrow 0}\frac{[(2+h)^{2}-(2+h)]-f(2)}{h}
RHD = \lim_{h\rightarrow 0}\frac{h^{2}+3h}{h} = 3
LHD = RHD
\frac{d|x^{2}-x|}{dx} = \frac{d\sqrt{(x^{2}-x)^{2}}}{dx}
Simply apply the chain rule, we have
\frac{df(x)}{dx} = \frac{x(2x^{2}-3x+1)}{|x(x-1)|}
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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