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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: Yoda on June 06, 2014, 10:45:05 pm

Title: modulus derivative?
Post by: Yoda on June 06, 2014, 10:45:05 pm: Differentiate 3|x|-x^3 and express it as a hybrid function over a suitable domain.
Title: Re: modulus derivative?
Post by: b^3 on June 06, 2014, 10:57:33 pm: Since the modulus is only applied to $x$ and not the whole function, we do the domain split about $x=0$ .
$\begin{alignedat}{1}f\left(x\right) & =3|x|-x^{3} \\ f\left(x\right) & =\begin{cases} 3\left(-x\right)-x^{3} & \text{If }x<0 \\ 3x-x^{3} & \text{If }x\geq0 \end{cases} \\ f'\left(x\right) & =\begin{cases} -3-3x^{2} & \text{ If }x<0 \\ 3-3x^{2} & \text{If }x>0 \end{cases} \end{alignedat}$

Note that $x=0$ is not included, as we have a 'sharp' point at $x=0$ at the ends of where the domain is split. That is the gradient of the two curves is not equal at $x=0$ .
https://www.desmos.com/calculator/bb2zodfhyb
Title: Re: modulus derivative?
Post by: Champ101 on June 06, 2014, 11:02:41 pm: Quote from: Yoda on June 06, 2014, 10:45:05 pm
Differentiate 3|x|-x^3 and express it as a hybrid function over a suitable domain.