Let |f (x)| be an absolute value function. Then the formula to find the derivative of |f (x)| is given below. Based on the formula given, let us find the derivative of |x|. |x|' = (ˣ⁄|ₓ|)(x)' |x|' = (ˣ⁄|ₓ|)(1) |x|' = ˣ⁄|ₓ|. Therefore, the derivative of |x| is ˣ⁄|ₓ|. Let y = |x|'. Then, we have y = ˣ⁄|ₓ|.

Tutorial on how to find derivatives of functions in calculus (Differentiation) involving the absolute value. A video on How to Find the derivative of an Absolute Value Function? is included. Let f(x) = |u(x)| f (x) = | u (x) |. Use the chain rule of differentiation to find the derivative of f = |u(x)| = u2(x)− −−−−√ f = | u (x) | = u 2 (x).

Nov 14, 2024 · Let $\size x$ be the absolute value of $x$ for real $x$. Then: $\dfrac \d {\d x} \size x = \dfrac x {\size x}$ for $x \ne 0$. Corollary. Let $u$ be a differentiable real function of $x$. Then: $\dfrac \d {\d x} \size u = \dfrac u {\size u} \dfrac {\d u} {\d x}$ for $u \ne 0$. At $u = 0$, $\size u$ is not differentiable. Proof

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Jun 21, 2017 · $\begingroup$ The absolute value function has a derivative(s) on restricted domains. i.e. f'(x) = -1 for x <0 and f'(x) = 1 for x > 0. However, the absolute value function is not "smooth" at x = 0 so the derivative at that point does not exist. $\endgroup$

Jul 2, 2019 · A derivative will measure the depth of the graph of a function at a random point on the graph. Therefore, the chosen derivative is called a slope. The derivative has a ratio of change in the function value to adjustment in the free variable.

Sep 11, 2024 · The absolute value function is not differentiable at x = 0 because the function has a sharp corner at that point, resulting in a discontinuity in the slope of the tangent lines. Let's discuss this in detail.

Aug 18, 2023 · The derivative of absolute value (function) is defined as the rate of change or the slope of a function at a specific point. The absolute value function is defined as: \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Apr 22, 2025 · The derivative of the absolute value function is defined piecewise as: \[ \frac{d}{dx} |x| = \begin{cases} 1 & \text{if } x > 0 \\[0.5em] -1 & \text{if } x < 0 \end{cases} \] The derivative does not exist at \( x = 0 \), because the function has a sharp corner at that point.

Since an absolute value function is represented by the graph of two “linear” equations coming together to form a “V” the derivative is a piecewise function of two CONSTANT values(represented by the slopes of those linear pieces).