Question: Let D stand for $\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathit{D}}^{\mathbf{2}}$, for $\frac{{\mathbf{d}}^{\mathbf{2}}}{{\mathbf{dx}}^{\mathbf{2}}},{\mathit{D}}^{\mathbf{3}}\mathbf{=}\frac{{\mathbf{d}}^{\mathbf{3}}}{{\mathbf{dx}}^{\mathbf{3}}}$, and so on. Are $\mathit{D}\mathbf{,}{\mathit{D}}^{\mathbf{2}}\mathbf{,}{\mathit{D}}^{\mathbf{3}}$ linear? Operate on functions of x which can be differentiated as many times as needed.

An operator is said to be a linear operator if the following relations are satisfied if $O(A+B)=O\left(A\right)+O\left(B\right)$ and,$O\left(kA\right)=kO\left(A\right)$ where, k is a number, and A and B, are numbers, functions, vectors, etc.

The differential operators.

$$\begin{gathered} D=\frac{d}{dx} \\ {D}^2=\frac{d^2}{d{x}^2} \\ {D}^3=\frac{d^3}{d{x}^3} \\ ⋮ \\ {D}^n=\frac{d^n}{d{x}^n} \end{gathered}$$

Which operates on the function f(x) by differentiating it with respect to x by n times represent a linear operator.

Find the differential $\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]$.

$\begin{array}{rcl}\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]\\ \frac{d}{dx}\left[kf\left(x\right)\right]& =& k\frac{d}{dx}\left[f\left(x\right)\right]\end{array}$

Where, k is a constant.

Find the differential $D^2\left[f\left(x\right)+g\left(x\right)\right]$.

$\begin{array}{rcl}{D}^2\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]\\ & =& \frac{d}{dx}\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]\\ & =& \frac{d}{dx}\left\{\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]\right\}\end{array}$

Solve further.

$\begin{array}{rcl}{D}^2\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d}{dx}\frac{df\left(x\right)}{dx}+\frac{d}{dx}\frac{dg\left(x\right)}{dx}\\ & =& \frac{d^2}{d{x}^2}f\left(x\right)+\frac{d^2}{d{x}^2}g\left(x\right)\\ & =& D^2\left[f\left(x\right)\right]+D^2\left[g\left(x\right)\right]\end{array}$

Find, $D^2\left[kf\left(x\right)\right]$.

$\begin{array}{rcl}{D}^2\left[kf\left(x\right)\right]& =& \frac{d^2}{d{x}^2}\left[kf\left(x\right)\right]\\ & =& k\frac{d^2}{d{x}^2}\left[f\left(x\right)\right]\\ & =& k{D}^2\left[f\left(x\right)\right]\end{array}$

Where, k is a constant

Find the differential $D^3\left[f\left(x\right)+g\left(x\right)\right]$.

$\begin{array}{rcl}{D}^3\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d^3}{d{x}^3}\left[f\left(x\right)+g\left(x\right)\right]\\ & =& \frac{d}{dx}\frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]\\ & =& \frac{d}{dx}\frac{d}{dx}\left\{\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]\right\}\\ & =& \frac{d}{dx}\left[\frac{d}{dx}\frac{df\left(x\right)}{dx}+\frac{d}{dx}\frac{dg\left(x\right)}{dx}\right]\end{array}$

Solve further,

$\begin{array}{rcl}{D}^3\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d}{dx}\left[\frac{d^2}{d{x}^2}f\left(x\right)+\frac{d^2}{d{x}^2}g\left(x\right)\right]\\ & =& \frac{d^3}{d{x}^3}f\left(x\right)+\frac{d^3}{d{x}^3}g\left(x\right)\\ & =& D^3\left[f\left(x\right)\right]+D^3\left[g\left(x\right)\right]\end{array}$

Find $D^3\left[kf\left(x\right)\right]$.

$\begin{array}{rcl}{D}^3\left[kf\left(x\right)\right]& =& \frac{d^3}{d{x}^3}\left[kf\left(x\right)\right]\\ & =& k\frac{d^3}{d{x}^3}\left[f\left(x\right)\right]\\ & =& k{D}^3\left[f\left(x\right)\right]\end{array}$

Where, k is a constant

Find the differential $D^n\left[f\left(x\right)+g\left(x\right)\right]$.

$\begin{array}{rcl}{D}^n\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d^n}{d{x}^n}\left[f\left(x\right)+g\left(x\right)\right]\\ & =& \frac{d}{dx}\frac{d}{dx}L\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]\\ & & \underset{ntimes}{⟷}\\ & =& \frac{d^n}{d{x}^n}\left[f\left(x\right)+g\left(x\right)\right]\end{array}$

Solve further,

$\begin{array}{rcl}{D}^n\left[f\left(x\right)+g\left(x\right)\right]& =& \frac{d^n}{d{x}^n}f\left(x\right)+\frac{d^n}{d{x}^n}g\left(x\right)\\ & =& D^n\left[f\left(x\right)\right]+D^n\left[g\left(x\right)\right]\end{array}$

Find $D^n\left[kf\left(x\right)\right]$.

$\begin{array}{rcl}{D}^n\left[kf\left(x\right)\right]& =& \frac{d^n}{d{x}^n}\left[kf\left(x\right)\right]\\ & =& k\frac{d^n}{d{x}^n}\left[f\left(x\right)\right]\\ & =& k{D}^n\left[f\left(x\right)\right]\end{array}$

Where, k is a constant.

Therefore, it has been shown that Differentiating operator $D^n$with the objects being operated on are the function f(x) by differentiating it with respect to x by n times represent a linear operator.