Question

The derivative of \[f\left( x \right) = x\left| x \right|\] is
A. \[2x\]
B. \[ - 2x\]
C. \[2{x^2}\]
D. \[2\left| x \right|\]

Answer 1

Hint: We know that the derivative of a function \[f\left( x \right)\] is the function whose value at \[x\] is \[f'\left( x \right)\]. So, we take the derivative of the given function with the help of the modulus function formula which gives a positive value when the function is greater than or equal to zero and a negative value when the function is lower than zero.


Formula Used:
The modulus function formula is
 \[\begin{array}{c}f\left( x \right) = \left| x \right|\\ = \left\{ \begin{array}{l}x,if\,x \ge 0\\ - x,if\,x < 0\end{array} \right.\end{array}\]

Complete step-by-step solution:
We are given that \[f\left( x \right) = x\left| x \right|\]
Now apply the modulus function formula, we get
\[\begin{array}{c}f\left( x \right) = x\left| x \right|\\ = \left\{ \begin{array}{l}x \cdot x = {x^2},if\,x \ge 0\\x \cdot \left( { - x} \right) = - {x^2},if\,x < 0\end{array} \right.\end{array}\]
Now we take the derivative of the function, and we get
\[f^{'}\left( x \right) = \left\{ \begin{array}{l}2x,x \ge 0\\ - 2x,x < 0\end{array} \right.\]
Therefore, \[f^{'}\left( x \right) = 2\left| x \right|\]
So, option (4) is correct
Hence, the derivative of the function \[f\left( x \right) = x\left| x \right|\] is \[2\left| x \right|\].

Additional information: In mathematics, the modulus of a real number \[x\]is given by the modulus function, denoted by \[\left| x \right|\]. It gives the non-negative value of \[x\]. The modulus or absolute value of a number is also considered as the distance of the number from the origin or zero. In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.