Question

What is the difference between $delta\left( x \right)$ and $dx$ ?

Answer 1

Hint: Here in this question we have been asked to write the difference between $delta\left( x \right)$ and $dx$. For answering this question we will be defining the terms and will give some examples and list out the differences.

Complete step-by-step solution:
Now considering from the question we have been asked to write the difference between $delta\left( x \right)$ and $dx$.
From the basic concepts we know that, the $delta\left( x \right)=\delta x$ is generally used to represent a small increment in the value of $x$ whereas the $dx$ is defined from the definition of the derivative $\displaystyle \lim_{\delta x \to 0}\dfrac{f\left( x+\delta x \right)-f\left( x \right)}{\delta x}=\dfrac{d}{dx}f\left( x \right)$ where $dx$ is actually a part of the operator.
Let us consider an example $f\left( x \right)={{x}^{2}}$ the derivative of this function will be given as $\begin{align}
  & \dfrac{d}{dx}f\left( x \right)\Rightarrow \displaystyle \lim_{h \to 0}\dfrac{{{\left( x+h \right)}^{2}}-{{x}^{2}}}{h} \\
 & =\displaystyle \lim_{h \to 0}\left( 2x+h \right)\Rightarrow 2x \\
\end{align}$
where $h$ is the small difference of $x$ that is $\delta x$ .
Let us consider another example where the value of $x$ is 12 initially and it changes to $12.0001$ which is almost ignorable change so the $\delta x$ is given as $0.0001$ .
Therefore we can conclude that the $delta\left( x \right)$ is the small change in the quantity $x$ whereas $dx$ is an operator used in integrations and differentiations.

Note: While answering questions of this type we should be sure with the concepts that we are going to use in between the steps because this is purely a concept based question and can be answered directly. Mistakes are possible only if we misunderstand the concepts. So through practice of the concepts can help in answering questions of this type.