# Derivative

• Apr 5th 2009, 12:45 PM
ment2byours
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. . .
• Apr 5th 2009, 12:59 PM
skeeter
Quote:

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. . .

$\frac{d}{dx}(\sin|x|)^3 = 3(\sin|x|)^2 \cdot (\cos|x|) \cdot \frac{x}{|x|}$

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

if $x < 0$ , $|x| = -x$
• Apr 5th 2009, 01:04 PM
Jhevon
alternatively, using the definition of $|x|$, you could realize you have the function:
$
f(x) = \left \{ \begin{array}{ll} (\sin x)^3 & \text{ if } x > 0 \\ (\sin (-x))^3 & \text{ if } x < 0 \end{array} \right.$

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

now differentiate piece by piece