Question

What is differentiation in simple words?

Answer 1

Differentiation is the rate of change of a quantity with respect to another quantity on which the first quantity depends on. There will be a dependent variable and an independent variable. There are first derivatives, second derivatives and so on of a function according to how many times we differentiate a function.

Complete step-by-step answer:
In Calculus, there is a terminology called differentiation which deals with some independent quantities and their dependent quantities. Differentiation is a process by which we can measure the rate of change of some quantity with respect to another quantity. These rates we get after differentiation are called derivatives.
Suppose that we have a function $y=f\left( x \right).$ Now, this is a function which has an independent variable $x$ and a dependent variable $y.$ we can see that the variable $y$ completely depends on the variable $x.$ So, in this function $x$ is the independent variable and $y$ is the dependent variable. That is, the value of $y$ can be found if we know the definition of the function and the value of $x.$
We use $\dfrac{d}{dx}$ to denote the derivative. If we are given with a problem in which we are asked to find $\dfrac{dy}{dx}$ where $y=f\left( x \right),$ then this means that we need to find out the derivative of the function $y=f\left( x \right)$ with respect to $x.$ This also means that differentiate $y$ with respect to $x.$ This can also be written as ${y}'={f}'\left( x \right).$
Until now, what we are talking about is the first derivative. That is, we differentiated a quantity only once. But we can differentiate a quantity more than once as ${y}''=\dfrac{{{d}^{2}}y}{d{{x}^{2}}},{y}'''=\dfrac{{{d}^{3}}y}{d{{x}^{3}}}...$ These are, respectively, second derivative, third derivative and so on.

Note: The derivative of a constant is zero. The quantities velocity, acceleration, et cetera, are the examples of derivatives. Velocity is the rate of change of displacement with respect to time. And the rate of change of velocity with respect to time is the acceleration. Or we can say that the velocity is the first derivative of displacement with respect to time and the acceleration is the second derivative of displacement with respect to time.