Question

What does derivative of y with respect to x mean?

Answer 1

Hint: 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:
 In calculus we will study the three concepts namely, limits, derivatives and integration. Here the three concepts are interlinked to each other.
Suppose $$f$$ is a real function and $$c$$ is a point in its domain. The derivative of $$f$$ at $$c$$ is
defined by $$\underset{c\to 0}{lim}{\displaystyle \frac{f(c+h)-f(c)}{h}}$$ provided this limit exists. Derivative of $$f$$ at $$c$$ is denoted by $$f^{\prime }(x)$$ or $${\displaystyle \frac{d}{dx}}f(x)$$.
The process of finding derivatives of a function is called differentiation.
Suppose $$f$$ is a function of x and usually it is denoted by $$y=f(x)$$
The derivative of y with respect to x is written by using the description which is present above as
$${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{d}{dx}}f(x)=f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$$
This is one way of representing.
If the function is a composite function then we use the concept of chain rule.
Let $$f$$ be a real valued function which is a composite of two functions u and v, it is represented as $$f=v\circ u$$. Suppose $$t=u(x)$$ and if both $${\displaystyle \frac{dt}{dx}}$$ and $${\displaystyle \frac{dv}{dt}}$$ exist, we have $${\displaystyle \frac{d}{dx}}f(x)={\displaystyle \frac{dv}{dt}}.{\displaystyle \frac{dt}{dx}}$$
Suppose $$f$$ is a function of x and usually it is denoted by $$y=f(x)$$
The derivative of y with respect to x is written by using the description which is present above as
$${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{d}{dx}}f(x)=f^{\prime }(x)={\displaystyle \frac{dy}{du}}.{\displaystyle \frac{du}{dx}}$$
This is another way of representing.

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 way of writing the derivative form. The derivative means rate of change of some quantity. One is normal one and the other one is chain rule concept.