Question

What is Leibniz Notation?

Answer 1

Hint: For solving this question you should know about the Leibniz notation. According to Leibniz notation we use the symbols $dx$ and $dy$ to represent the infinitely small increments of $x$ and $y$. As the $\Delta x$ and $\Delta y$ represent the finite increments of $x$ and $y$ respectively. We can write the first and second derivative as $\dfrac{dy}{dx}$ and $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ with respect to $x$ in Leibniz notation.

Complete step by step solution:
According to the question we have to explain the Leibniz notation. The Leibniz notation is used to represent the infinitely small increments of $x$ and $y$. If we look up an example for it then it will be clear exactly.
Example: What is the derivative of $y$ with respect to $x$, given that: $4{{y}^{2}}+8y={{x}^{2}}$?
In this question if we think then we have to calculate the $\dfrac{dy}{dx}$, then in the first look it will look like a hard question but it is very easy. If we take the derivatives from here, then,
$\begin{align}
  & \Rightarrow \dfrac{d}{dx}\left( 4{{y}^{2}}+8y \right)=\dfrac{d}{dx}\left( {{x}^{2}} \right) \\
 & \Rightarrow 4\dfrac{d}{dx}\left( {{y}^{2}} \right)+8\dfrac{d}{dx}\left( y \right)=2x \\
\end{align}$
Now, again $y$ is the function of $x$, so we use the chain rule for any derivatives involving $y$. So, we get,
$\begin{align}
  & 4\dfrac{d}{dx}\left( {{y}^{2}} \right)+8\dfrac{d}{dx}\left( y \right)=2x \\
 & \Rightarrow 4.2y\dfrac{dy}{dx}+8\dfrac{dy}{dx}=2x \\
 & \Rightarrow 8y.\dfrac{dy}{dx}+8\dfrac{dy}{dx}=2x \\
\end{align}$
Now, take $\dfrac{dy}{dx}$ common from L.H.S, we get,
$\begin{align}
  & \dfrac{dy}{dx}\left( 8y+8 \right)=2x \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{2x}{8y+8} \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{x}{4\left( y+1 \right)} \\
\end{align}$
So, here the differentiation of $4{{y}^{2}}+8y={{x}^{2}}$ is done. And here $\dfrac{dy}{dx}$ is the Leibniz notation which is the first derivative of the function.
In the Leibniz notation which we derive from the use of a capital letter $\Delta $ to indicate the finite increments in variable quantity. If the function $f$ is differentiable at $x$ and we set $y=f\left( x \right)$, then the derivative $f'\left( x \right)$ is defined as $f'\left( x \right)=\dfrac{dy}{dx}=\displaystyle \lim_{\Delta x \to 0}\dfrac{\Delta y}{\Delta x}=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$.
Thus, Leibniz notation is explained.

Note: During solving any problem of differentiation we always use $\Delta $ if the increments are infinite. But if the increment is very small or we can say that the limit of $\Delta $ or $\Delta x$ goes to zero, then it will be represented by $d$ or $dx$. And it is the form of Leibniz notation.