1. Derivitive Problem

Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

a. Find f '(x) for x>0
b. Find f '(x) for x<0
c. Determine whether f9s0 is continuous x = 0. Justify your answer.
d. Determine whether the derivative of f(x) exists at x=0. Justify your answer

Thanks a lot for the help, I had a lot of trouble throwing the absolute value into the derivative.

-Aussiekid90

2. Originally Posted by aussiekid90
Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

a. Find f '(x) for x>0
b. Find f '(x) for x<0
c. Determine whether f9s0 is continuous x = 0. Justify your answer.
d. Determine whether the derivative of f(x) exists at x=0. Justify your answer

Thanks a lot for the help, I had a lot of trouble throwing the absolute value into the derivative.

-Aussiekid90
Hint: What is the value of f(x) when x is negative? What is the value of f(x) when x is positive? Then construct f(x) as a piece-wise function and take the derivative of each piece. (Note: f'(x) is NOT defined at x = 0!)

-Dan

3. Originally Posted by aussiekid90
Let f be the real-valued function defined by f(x)=sin^3(x)+sin^3(|x|)

a. Find f '(x) for x>0
b. Find f '(x) for x<0
c. Determine whether f9s0 is continuous x = 0. Justify your answer.
d. Determine whether the derivative of f(x) exists at x=0. Justify your answer
For $x>0$ we have,
$f=\sin^3x+\sin^3x=2\sin^3x$
Then,
$f'=6\cos x \sin^2 x$
Thus, the derivative from the right at x=0 is,
$f'(0)^+=6\cos (0) \sin^2 (0)=0$

For $x<0$ we have,
$f=\sin^3x-\sin^3x=0$
Then,
$f'=0$
Thus, the derivative from the left at x=0 is,
$f'(0)^-=0$
Thus, the function is indeed differencial at $x=0$. Indeed everywhere.
Which means the function is continous everywhere.