# What is the derivative of the absolute value of cos(x)?

1. Oct 22, 2005

### math&science

What is the derivative of the absolute value of cos(x)?

2. Oct 22, 2005

### TD

The derivative of cos(x) is -sin(x) and the derivative of |x| is sgn(x), can you now combine them?

3. Oct 22, 2005

### math&science

Thanks, but what does sgn stand for? Is the derivative just -sin(x)*Abs(cos(x))'?

Last edited: Oct 22, 2005
4. Oct 22, 2005

### TD

The 'sign' or 'signum' function, which returns 1 or -1, whether the argument in question was positive or negative.

See Mathworld or Wikipedia.

5. Oct 22, 2005

### benorin

$$\frac{d}{dx}|\cos(x)|=-\frac{|\cos(x)|}{\cos(x)}\sin(x)$$
Note that the signum function can be defined by $$sgn(x)=\frac{|x|}{x}$$ for nonzero x, and is zero when x is zero. The signum function cannot be use in this case as $$|\cos(x)|$$ is not differentiable at the values of x for which $$\cos(x)=0$$ as the lefthand and righthand derivative are not equal there (by lefthand or righthand derivates, what is meant is the left or right-handed limit of the difference quotient at a particular value of x).

6. Oct 22, 2005

### math&science

Thank you so much. I've never even heard about the signum function before until now. How would I go about taking higher order derivatives of the signum function like the second and third, etc. How does that work?

7. Oct 22, 2005

### hypermorphism

Look at its graph. The derivative should be apparent.