Question

Explain the procedure to find $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ (second order derivative) of any function $y=f\left( x \right)$

Answer 1

In this problem we need to explain the procedure to find the second order derivative of any function $y=f\left( x \right)$. For this we will first explain what is the second order derivative means and what are the requirements to calculate the second order derivative. After that we will assume any function and we will calculate the second order derivative of the assumed function.

Complete step-by-step answer:
In the problem we can observe the term second order derivative. The name itself suggests that we need to do a particular task two time that means the second order derivative of a function is nothing but the value when we differentiated the given function two times. Mathematically we can write it as
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}\left( \dfrac{d}{dx}\left( y \right) \right)$
From the above equation we can say that in order to calculate the second order derivative we must need to the value of first order derivative which is $\dfrac{d}{dx}\left( y \right)\text{ or }\dfrac{dy}{dx}$. So, while calculating the second order derivative we need to first calculate the first order derivative.
Let us assume a function $y=2{{x}^{2}}+3$.
In order to calculate the second order derivative of the above assumed function we need to have it first order derivative value as we have discussed earlier. So, differentiating the assumed function with respect to $x$, then we will get
$\begin{aligned} & \dfrac{dy}{dx}=\dfrac{d}{dx}\left( 2{{x}^{2}} \right)+\dfrac{d}{dx}\left( 3 \right) \\\ & \Rightarrow \dfrac{dy}{dx}=2\left( 2x \right)+0 \\\ & \Rightarrow \dfrac{dy}{dx}=4x \\\ \end{aligned}$
We have the first order derivative of the assumed function as $\dfrac{dy}{dx}=4x$.
We are going to again differentiate the above value to find the second order derivative, then we will have
$\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)=\dfrac{d}{dx}\left( 4x \right)$
We know that $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}\left( \dfrac{d}{dx}\left( y \right) \right)$. Substituting this value in the above equation, then we will get
$\begin{aligned} & \dfrac{{{d}^{2}}y}{d{{x}^{2}}}=4\left( 1 \right) \\\ & \Rightarrow \dfrac{{{d}^{2}}y}{d{{x}^{2}}}=4 \\\ \end{aligned}$

Hence the second order derivative of the assumed function $y=2{{x}^{2}}+3$ is $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=4$.

Note: In this problem they have asked about second order derivative only. We have third order, fourth order and so ${{n}^{th}}$ order derivatives for a function. For all these derivatives we can follow the above procedure which is differentiating the given function as many times that is equal to the given order.