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

Question

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

zarkon · Answer

use the definition of the derivative

anonymous · Answer

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

zarkon · Answer

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

anonymous · Answer

then it's just 0?

anonymous · Answer

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

anonymous · Answer

actually, it would be a horizontal slope.

anonymous · Answer

which would be 0?

zarkon · Answer

attachment

anonymous · Answer

Haha. Ok. Thanks!

zarkon · Answer

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