Question

Calculate f '(0) for the following continuous function. (If an answer does not exist, enter DNE.)
f(x) = x|x|

Answer 1

use the definition of the derivative

Answer 2

I'm just not sure how to do that with an absolute value.

Answer 3

\[f'(0)=\lim_{h\to 0}\frac{f(0+h)-f(0)}{h}\]

\[f'(0)=\lim_{h\to 0}\frac{(0+h)|0+h|-0|0|}{h}=\cdots\]

Answer 4

then it's just 0?

Answer 5

or does that derivative not exist, because at 0, there's a vertical slope of the tangent line in the function?

Answer 6

actually, it would be a horizontal slope.

Answer 7

which would be 0?

Answer 8

0

Answer 9

Haha. Ok. Thanks!

Answer 10

\[f'(0)=\lim_{h\to 0}\frac{(0+h)|0+h|-0|0|}{h}=\cdots\]

\[=\lim_{h\to 0}\frac{h|h|}{h}=\lim_{h\to 0}|h|=0\]