The derivative of an absolute value function is a function that describes the rate of change of the absolute value of a function with respect to a variable. The absolute value function is defined as the value of a number without regard to its sign, so the derivative of the absolute value of a function is the derivative of the function itself if the function is positive, and the negative of the derivative of the function if the function is negative.
To calculate the derivative of an absolute value function, we first need to determine the sign of the function at the point where we want to find the derivative. If the function is positive at that point, then the derivative of the absolute value function is the same as the derivative of the function itself. If the function is negative at that point, then the derivative of the absolute value function is the negative of the derivative of the function itself. Once we know the sign of the function at the point where we want to find the derivative, we can use the following formula to calculate the derivative of the absolute value function:
$$f'(x) = \begin{cases} f'(x) & \text{if } f(x) \ge 0 \\\ -f'(x) & \text{if } f(x) < 0 \end{cases}$$
Where $f(x)$ is the absolute value function.
How to Take Derivative of Absolute Value
The derivative of the absolute value function |x| is defined as follows:
if x > 0, then |x|’ = 1
if x < 0, then |x|’ = -1
if x = 0, then |x|’ = 0
In other words, the derivative of the absolute value function is equal to 1 if the input is positive, -1 if the input is negative, and 0 if the input is zero.
People Also Ask About How to Take Derivative of Absolute Value
How do you find the derivative of |x^2 – 1|?
Answer:
First, we need to find the derivative of x^2 – 1, which is 2x. Then, we need to apply the chain rule, which states that the derivative of |u(x)| is u'(x) * |u(x)|’ where u(x) = x^2 – 1. So, the derivative of |x^2 – 1| is 2x * |x^2 – 1|’ which equal to 2x * 1 because x^2 – 1 is always positive except 0.
How do you find the derivative of |x| + |y|?
Answer:
We can use the sum rule for derivatives, which states that the derivative of f(x) + g(x) is f'(x) + g'(x). So, the derivative of |x| + |y| is |x|’ + |y|’ which equal to 1 if x > 0 + 1 if y > 0 so it equal 2 if x and y > 0, 1 if just one of them bigger than 0 and 0 if both less than 0
