Question

Find the derivative of $\left| {2{x^2} - 3} \right|$ with respect to $x$ .

Answer 1

The sum rule and constant rule is applicable here. We need to differentiate the variable $2{x^2}$ and $3$ differently by using different rules applicable to each of them.

Formula used: The formulae used in the solution are given here.
$\dfrac{d}{{dx}}n{x^a} = na{x^{a - 1}}$ where $a$ and $n$ are real numbers.
By sum rule, $\left( {\alpha f + \beta g} \right)' = \alpha f' + \beta g'$ for all functions $f$ and $g$ and for all real numbers $\alpha$ and $\beta$ .
By constant rule, if $f\left( x \right)$ is a constant, then its derivative is $f'\left( x \right) = 0.$

Complete Step by Step Solution
The function given here is $f\left( x \right) = \left| {2{x^2} - 3} \right|$ .
Differentiation is the action of computing a derivative. The derivative of a function $y = f\left( x \right)$ of a variable $x$ is a measure of the rate at which the value $y$ of the function changes with respect to the change of the variable $x$ . It is called the derivative of $f$ with respect to $x$ .
If $x$ and $y$ are real numbers, and if the graph of $f$ is plotted against $x$ , the derivative is the slope of this graph at each point.
To find the derivative, we differentiate $f\left( x \right)$ with respect to $x$ .
$f'\left( x \right) = \dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}} = \dfrac{d}{{dx}}\left( {2{x^2} - 3} \right)$
According to the sum rule, which states that, $\left( {\alpha f + \beta g} \right)' = \alpha f' + \beta g'$ for all functions $f$ and $g$ and for all real numbers $\alpha$ and $\beta$ .
We can write this in a simpler way as, $\dfrac{d}{{dx}}\left( {2{x^2}} \right) - \dfrac{d}{{dx}}\left( 3 \right) = 4x$ .
This is according to, $\dfrac{d}{{dx}}n{x^a} = na{x^{a - 1}}$ where $a$ and $n$ are real numbers.
By constant rule, if $f\left( x \right)$ is a constant, then its derivative is $f'\left( x \right) = 0.$
Thus, the value of $\dfrac{d}{{dx}}\left( 3 \right)$ is zero, since $3$ is a constant.

Note
Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives.