Derivative of Work

We now understand the derivative of work. It is commonly represented as dW/dt, where W is work, and its value depends on the context (such as force, distance, or time). The concept of work has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Work: Work is defined as the force applied over a distance (W = F·d).

Force: Force is any interaction that, when unopposed, will change the motion of an object.

Distance: The amount of space between two points, which is part of the work equation.

The derivative of work can be denoted as dW/dt. The formula we use to differentiate work depends on the specific context: dW/dt = F·d' + F'·d

The formula applies to all situations where force and distance are functions of time and are differentiable.

We can derive the derivative of work using proofs. To show this, we will use physical principles along with the rules of differentiation.

There are several methods we use to prove this, such as:

• By First Principle
   
• Using Chain Rule
   
• Using Product Rule

We will now demonstrate the differentiation of work using the above-mentioned methods:

By First Principle

The derivative of work can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of work using the first principle, consider W(t) = F(d(t)). Its derivative can be expressed as the following limit: W'(t) = limₕ→₀ [W(t + h) - W(t)] / h Given that W(t) = F(d(t)), we write W(t + h) = F(d(t + h)). Substituting these into the equation: W'(t) = limₕ→₀ [F(d(t + h)) - F(d(t))] / h = limₕ→₀ [F(d(t) + h) - F(d(t))] / h = F'(d(t))·d'(t) In this context, work is a function of force and distance, and the derivative reflects how work changes with these variables.

Using Chain Rule

To prove the differentiation of work using the chain rule, We use the formula: W = F(d) Consider F = F(t) and d = d(t), By chain rule: dW/dt = F'(t)·d(t) + F(t)·d'(t) This shows how the rate of change of work depends on the rate of change of force and distance.

Using Product Rule

We will now prove the derivative of work using the product rule. The step-by-step process is demonstrated below: Here, we use the formula: W = F·d Given that F = F(t) and d = d(t), Using the product rule formula: d/dt [F·d] = F'·d + F·d' This highlights how work's rate of change is affected by both force and distance over time.

When work is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like work.

For the first derivative of work, we write W′(t), which indicates how work changes over time. The second derivative is derived from the first derivative, which is denoted using W′′(t). Similarly, the third derivative, W′′′(t), is the result of the second derivative, and this pattern continues.

For the nth Derivative of work, we generally use W^(n)(t) for the nth derivative of a function W(t), which tells us the change in the rate of change, continuing for higher-order derivatives.

When force is constant, the derivative of work simplifies to dW/dt = F·d'.

When distance is constant, the derivative of work simplifies to dW/dt = F'·d.

When both force and distance change with time, use the full derivative formula.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in its variables.

• Product Rule: A rule used to differentiate functions that are products of two or more functions.

• Chain Rule: A rule used to differentiate composite functions.

• Work: Defined as the force applied over a distance, often used in physical equations.

• Force: A physical quantity that causes an object to undergo a change in speed, direction, or shape.