Question

Double differentiate of displacement with respect to time is
A. Acceleration
B. Velocity
C. Force
D. None.

Answer 1

We will first define what is a second order derivative. Then we will assume a variable for the displacement and differentiate it with respect to time and get the first order derivative and then we will again differentiate it to get the second order derivative and finally get the answer.

Complete step by step answer:
First let us understand what is a double differentiation, it is also called a second-order derivative. We know that the first derivative $\dfrac{dy}{dx}$ represents the rate of the change in $y$ with respect to $x$ . The second-order derivative is nothing but the derivative of the first derivative of the given function.
Mathematically, if $y=f\left( x \right)$ , then $\dfrac{dy}{dx}=f'\left( x \right)$
Now if $f'\left( x \right)$ is differentiable then differentiating $\dfrac{dy}{dx}$ again with respect to $x$ we will get the second order derivative that is: $\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)=\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=f''\left( x \right)$ .
So we know that displacement is the vector that specifies the change in position of a point, particle, or object. The position vector directed from the reference point to the present position.
Let’s the displacement be $x$ ,
Now we will start by differentiating with respect to time so we will get: $\dfrac{dx}{dt}$ , now this is the first derivative and we know that the velocity is defined as the rate of change of position or the rate of displacement, therefore first derivative of displacement is velocity.
Now to get the second order derivative we will again differentiate the first order derivative that is $\dfrac{dx}{dt}$ ,So we will get: $\dfrac{d}{dt}\left( \dfrac{dx}{dt} \right)=\dfrac{{{d}^{2}}x}{d{{t}^{2}}}$ , now we know that the acceleration is defined as the rate of change of velocity.

Therefore, the second order derivative of displacement is acceleration.
Hence, the correct option is A.

Note:
We can further derive the displacement like the third order derivative is jerk or jolt and similarly fourth derivative as jounce. Note that the first derivative represents the slope of the function at a point, and the second derivative describes how the slope changes over the independent variable in the graph. For a function having a variable slope, the second derivative explains the curvature of the given function.