Simplify: g(x)=(1-|x|)/(2-|x|)

Identify function components: Identify the function and its components. We have g(x) = (1-|x|)/(2-|x|). Here, the numerator is 1-|x| and the denominator is 2-|x|.Differentiate numerator and denominator: Differentiate the numerator and denominator. For the numerator, u(x) = 1-|x|, u'(x) = -sign(x) where sign(x) is the sign function. For the denominator, v(x) = 2-|x|, v'(x) = -sign(x).Apply quotient rule: Apply the quotient rule. The quotient rule is (v*u' - u*v') / v^2. Plugging in: u' = -sign(x), v' = -sign(x), u = 1-|x|, v = 2-|x|. g'(x) = ((2-|x|)*(-sign(x)) - (1-|x|)*(-sign(x))) / (2-|x|)^2.Simplify the derivative: Simplify the derivative. g'(x) = (-2*sign(x) + |x|*sign(x) + sign(x) - |x|*sign(x)) / (2-|x|)^2. g'(x) = (-2*sign(x) + sign(x)) / (2-|x|)^2. g'(x) = -sign(x) / (2-|x|)^2.

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Calculus

Find derivatives using the chain rule I

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Q. Simplify:

g(x)=\frac{1-|x|}{2-|x|}

Identify function components: Identify the function and its components. $\newline$ We have $g(x) = \frac{1-|x|}{2-|x|}$ . Here, the numerator is $1-|x|$ and the denominator is $2-|x|$ .
Differentiate numerator and denominator: Differentiate the numerator and denominator. $\newline$ For the numerator, $u(x) = 1-|x|$ , $u'(x) = -\text{sign}(x)$ where $\text{sign}(x)$ is the sign function. $\newline$ For the denominator, $v(x) = 2-|x|$ , $v'(x) = -\text{sign}(x)$ .
Apply quotient rule: Apply the quotient rule. $\newline$ The quotient rule is $(v*u' - u*v') / v^2$ . Plugging in: $\newline$ $u' = -\text{sign}(x)$ , $v' = -\text{sign}(x)$ , $u = 1-|x|$ , $v = 2-|x|$ . $\newline$ $g'(x) = ((2-|x|)*(-\text{sign}(x)) - (1-|x|)*(-\text{sign}(x))) / (2-|x|)^2$ .
Simplify the derivative: Simplify the derivative. $\newline$ $g'(x) = (-2\cdot\text{sign}(x) + |x|\cdot\text{sign}(x) + \text{sign}(x) - |x|\cdot\text{sign}(x)) / (2-|x|)^2.$ $\newline$ $g'(x) = (-2\cdot\text{sign}(x) + \text{sign}(x)) / (2-|x|)^2.$ $\newline$ $g'(x) = -\text{sign}(x) / (2-|x|)^2.$