Differentiation

The differentiation is to find the amount of change at a point on a graph f(x). This is like the process of finding the slope of the tangent line when you can draw a tangent at a point on a graph.

For example, if one point of coordinates (x, y) is changed to (x ‘, y’), then the slope of the equation is

\[ slope = \frac{{y}’ – y}{{x}’ – x} \]

In general, scientists write the Greek letter ‘Δ (delta)’ as the amount of change and write it as Δx, Δy. Δx is the amount of change in x. It equals to (x ‘-x).

\[ slope = \frac{ \Delta y}{\Delta x} \]

Differential in Physics

The differentiation is said to have been discovered by Newton and Leibniz respectively.

Newton introduced a differentiation to analyze the motion of objects.

An example of the differentiation that is mainly dealt with in classical mechanics is as follows.

The derivative of position with time is velocity. \( v = \frac{\Delta s}{\Delta t} \)

The derivative of velocity with time is acceleration. \( a = \frac{\Delta v}{\Delta t} \)

The derivative of momentum with time is force. \( f = m \frac{\Delta v}{\Delta t} \)

A derivative of volume of a sphere(\( \frac{4}{3} \pi r^3 \)), with respect to the radius of a sphere, gives the surface area of a sphere(\( 4 \pi r^2 \)).