Question

What is the derivative of $f\left( x \right)={{x}^{3}}-3x$ ?

Answer 1

To find the derivative of the given function $f\left( x \right)={{x}^{3}}-3x$, we are going to use the following derivative form: $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$. And this derivative form we are going to apply first on ${{x}^{3}}$ and then on $x$. Also, we are going to use the derivative form which says: $\dfrac{d\left( kg\left( x \right) \right)}{dx}=k\dfrac{dg\left( x \right)}{dx}$.

Complete step-by-step solution:
The function given in the above problem which we are going to take the derivative is as follows:
$f\left( x \right)={{x}^{3}}-3x$
Now, taking derivative with respect to x on both the sides of the above equation we get,
$\dfrac{df\left( x \right)}{dx}=\dfrac{d\left( {{x}^{3}}-3x \right)}{dx}$
Distributing derivative amongst the two functions ${{x}^{3}}\And 3x$ we get,
$\dfrac{df\left( x \right)}{dx}=\dfrac{d{{x}^{3}}}{dx}-\dfrac{d\left( 3x \right)}{dx}$ ………………. (1)
Now, to differentiate the above function, we are going to use the following derivative property which states that:
$\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$
Applying the above property on $\dfrac{d{{x}^{3}}}{dx}$ by substituting the value of n as 3 in the above equation and we get,
$\begin{aligned} & \dfrac{d{{x}^{3}}}{dx}=3{{x}^{3-1}} \\\ & \Rightarrow \dfrac{d{{x}^{3}}}{dx}=3{{x}^{2}} \\\ \end{aligned}$
From the above, we have found the derivative of one of the part of the two derivatives so substituting the above value in eq. (1) we get,
$\dfrac{df\left( x \right)}{dx}=3{{x}^{2}}-\dfrac{d\left( 3x \right)}{dx}$ ………… (2)
Now, we are going to apply the following derivative form:
$\dfrac{d\left( kg\left( x \right) \right)}{dx}=k\dfrac{dg\left( x \right)}{dx}$
Substituting the value of k as 3 and $g\left( x \right)=x$ in the above equation we get,
$\dfrac{d\left( 3x \right)}{dx}=3\dfrac{dx}{dx}$
In the R.H.S of the above equation, $dx$ will get cancelled out from the numerator and the denominator and we get 1 in place of $\dfrac{dx}{dx}$ and we get,
$\begin{aligned} & \dfrac{d\left( 3x \right)}{dx}=3\left( 1 \right) \\\ & \Rightarrow \dfrac{d\left( 3x \right)}{dx}=3 \\\ \end{aligned}$
The above derivative is the second part of eq. (1) so using the above relation in eq. (2) we get,
$\dfrac{df\left( x \right)}{dx}=3{{x}^{2}}-3$
Now, taking 3 as common in the R.H.S of the above equation we get,
$\dfrac{df\left( x \right)}{dx}=3\left( {{x}^{2}}-1 \right)$
Hence, we have found the derivative of the given function as: $3\left( {{x}^{2}}-1 \right)$.

Note: To solve the above problem you should know the following derivative forms otherwise you could not solve the problem further:
$\begin{aligned} & \dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}; \\\ & \dfrac{d\left( kg\left( x \right) \right)}{dx}=k\dfrac{dg\left( x \right)}{dx} \\\ \end{aligned}$
So, make sure you have properly understood the above derivatives.