Derivative

The ratio of change in a varying quantity with change in another varying quantity is called a derivative.

Introduction

A differential is a difference of two quantities of a varying quantity. In calculus, two differentials are compared by calculating a ratio of them and it is useful to us for studying the change in one varying quantity in the point of view of change in another varying quantity. The value of the ratio is called a derivative in differential calculus.

$x$ and $y$ are two variables. The values of $x$ at two different points are $x_{1}$ and $x_{2}$ and the values of $y$ at two different points are $y_{1}$ and $y_{2}$.

Change in Variables

If, the change in variables $x$ and $y$ are not infinitely small, then the differential are denoted by $\Delta x$ and $\Delta y$ respectively.

$\Delta x \,=\, x_{2}-x_{1}$ and $\Delta y \,=\, y_{2}-y_{1}$

Now, compare both quantities by calculating the ratio of change in one varying quantity to change in another varying quantity. In this example, the ratio of $\Delta y$ to $\Delta x$ is calculated and the value of the ratio of them is called a derivative.

$\dfrac{\Delta y}{\Delta x}$

The mathematical expressions states that the comparison of change in one varying quantity with respect to change in another varying quantity. Hence, it is read as derivative of $y$ with respect to $x$.

Infinitesimal Change in Variables

If the change in variables $x$ and $y$ are infinitely small, then the differentials are written as $dy$ and $dx$ respectively in differential calculus.

$dx \,=\, x_{2}-x_{1}$ and $dy \,=\, y_{2}-y_{1}$

Calculate ratio of quantities to get the value of them and it is known as derivative.

$\dfrac{dy}{dx}$

It is read as derivative of $y$ with respect to $x$

Formula

The definition of the derivative is popularly expressed in mathematical form in two ways. Remember, The variable $y$ is generally used to represent the function $f{(x)}$ in calculus.

$(1) \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{f{(x+\Delta x)}-f{(x)}}{\Delta x}}$

$(2) \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}}{h}}$

In this formula, $\Delta x$ is simply represented by $h$ because they both are same.

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