Question

What is the derivative of an absolute value?

Answer 1

Hint: The derivative is defined for a function which is the differentiation of the dependent variable with respect to the independent variable. So we have to write the function for the absolute value of a variable x, and then differentiate it with respect to x to get the answer. So we will let a function $y=\left| x \right|$

Complete step-by-step solution:
We know that the absolute value of a variable x is defined as the non-negative value of x. It is also known as the modulus of x. So let us assume a function y as
$\Rightarrow y=\left| x \right|......\left( i \right)$
Now, we know that the derivative of a function is defined as the differentiation of the dependent variable with respect to the independent variable. So we need to differentiate y with respect to x for obtaining the derivative for the absolute value. For that, we have to simplify the expression of modulus of x.
We know that the square root of a number is always non-negative, and so is the absolute value. This means that the modulus of x must be equal to the square root of the square of x. Therefore, the equation (i) can also be written as
$\begin{align}
  & \Rightarrow y=\sqrt{{{x}^{2}}}.......\left( ii \right) \\
 & \Rightarrow y={{\left( {{x}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}$
Differentiating both the sides with respect to x, we get
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( {{\left( {{x}^{2}} \right)}^{\dfrac{1}{2}}} \right)}{dx}$
On applying the chain rule for differentiation, we get
\[\begin{align}
  & \Rightarrow \dfrac{dy}{dx}=\dfrac{1}{2}{{\left( {{x}^{2}} \right)}^{\dfrac{1}{2}-1}}\dfrac{d\left( {{x}^{2}} \right)}{dx} \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{1}{2}{{\left( {{x}^{2}} \right)}^{-\dfrac{1}{2}}}\left( 2x \right) \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{x}{{{\left( {{x}^{2}} \right)}^{\dfrac{1}{2}}}} \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{x}{\sqrt{{{x}^{2}}}} \\
\end{align}\]
From (ii) we can right the denominator of the RHS as
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{x}{y}\]
Putting the equation (i) in the above equation, we finally get
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{x}{\left| x \right|}\]
Hence, the derivative of the absolute value of x is equal to \[\dfrac{x}{\left| x \right|}\].

Note: The derivative of the absolute value of x takes the value \[1\] for \[x>0\], and $-1$ for $x<0$. From the expression for the derivative of the absolute value \[\dfrac{x}{\left| x \right|}\], we can observe that it will not be defined at $x=0$ as that would give us $\dfrac{0}{0}$ which is not defined.