differentials

The derivative method is a method of obtaining the amount of change at a point of a graph f(x). This is similar to the process of finding the tangent slope at a point in a graph.

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

\[slope\,=\, \frac { y’ -y }{ x’-x } \, = \, \frac{Δy}{Δx} \]

Δx is “variation of x” and is equal to (x’ – x).

Δy is “variation of y” and is equal to (y’ – y).

differentials in Classical Mechanics

Differentials are said to have been discovered by Newton and Leibniz respectively.

Newton introduced Differentials to analyze the motion of objects.

Examples of differentials that are mainly covered in classical mechanics are:

- Velocity, \(V(t)\) is the derivative of position. \( v\,=\,\frac{Δs}{Δt} \)
- Acceleration, \(A(t)\), is the derivative of velocity. \( a\,=\,\frac{Δv}{Δt} \)
- Force, \(F(t)\), is the derivative of momentum. \( f\,=\,m\frac{Δv}{Δt} \)
- Area of sphere( \( 4\pi r^2 \) ) is obtained by differentiating the volume( \( \frac{4}{3}\pi r^3 \) ) with radius r.