1. ## Derivative

What is the derivative of (sin|x|)^3?

-for x>0
-for x<0

I think it is 3sin|x|*(cos|x|) for the first and 3sin|x|*(-cos|x|) for the second but I'm not sure. . .

2. Originally Posted by ment2byours
What is the derivative of (sin|x|)^3?

-for x>0
-for x<0

I think it is 3sin|x|*(cos|x|) for the first and 3sin|x|*(-cos|x|) for the second but I'm not sure. . .
$\displaystyle \frac{d}{dx}(\sin|x|)^3 = 3(\sin|x|)^2 \cdot (\cos|x|) \cdot \frac{x}{|x|}$

if $\displaystyle x > 0$ , $\displaystyle |x| = x$

if $\displaystyle x < 0$ , $\displaystyle |x| = -x$

3. alternatively, using the definition of $\displaystyle |x|$, you could realize you have the function:
$\displaystyle f(x) = \left \{ \begin{array}{ll} (\sin x)^3 & \text{ if } x > 0 \\ (\sin (-x))^3 & \text{ if } x < 0 \end{array} \right.$

$\displaystyle {\color{white}f(x)} = \left \{ \begin{array}{ll} (\sin x)^3 & \text{ if } x > 0 \\ -(\sin x)^3 & \text{ if } x < 0 \end{array} \right.$

now differentiate piece by piece