Resolved: "I was just wondering what the derivative of..."

khi1la2f1qv

Answered question

2021-11-12

I was just wondering what the derivative of

f (x) = max {(0, 1 - x)}^{2}

would be. What technique do you use to determine this derivative?

Melinda Olson

Beginner2021-11-13Added 20 answers

It might be of help to sketch the function or write it without the max. We get

f (x) = {\begin{cases} (1 - x)^{2} & i f x \leq 1 \\ 0 & i f x \geq 1 \end{cases}

It is easy to work out the derivative everywhere except at x=1. At x=1, work out explicitly from definition.

lim_{h \to 0^{+}} \frac{f (1 + h) - f (1)}{h} = 0

lim_{h \to 0^{-}} \frac{f (1 + h) - f (1)}{h} = lim_{h \to 0^{-}} \frac{h^{2}}{h} = 0

Hence, we have

f (x) = {\begin{cases} 2 (x - 1) & i f x \leq 1 \\ 0 & i f x \geq 1 \end{cases}

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