Question

What is the formal definition of derivative?

Answer 1

Here for this question the solution will be in the form of a descriptive way. So here we explain the concept of derivative and how we will write the derivative for the given function. So to solve this problem we must know the concept of differentiation and derivative.

Complete step-by-step solution:
The derivative of a function is one of the basic concepts of mathematics which comes under the concept of calculus. The process of finding the derivative is called differentiation. The inverse for differentiation is called integration.
The derivative of a function at some point represents the rate of change of the function at this point. We can estimate the rate of change by calculating the ratio of change of the function $\Delta y$ to the change of the independent variable $\Delta x$. In the definition of derivative, this ratio is considered in the limit as $\Delta x \to 0$.
The definition of the derivative is usually defined by
Let $f(x)$ be a function whose domain contains an open interval about some point ${x_0}$. Then the function $f(x)$ is said to be differentiable at ${x_0}$, and the derivative of $f(x)$ at ${x_0}$ is given by
$f'({x_0}) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{f({x_0} + \Delta x) - f({x_0})}}{{\Delta x}}$
To represents the derivative, we use two different notations forms and then we represents and that is
Lagrange’s notation is to write the derivative of the function $y = f(x)$ as $f'(x)$ or $y'(x)$
Leibniz’s notation is to write the derivative of the function $y = f(x)$ as $\dfrac{{df}}{{dx}}$ or $\dfrac{{dy}}{{dx}}$.
Hence this the general description of the derivative, where y is the dependent variable and x is the independent variable.

Note: The y represents the dependent function where y is dependent on the value of x. In the question they have mentioned to describe the definition of the derivatives the derivative means rate of change of some quantity. This one is the descriptive form and which is related to the derivative concept.