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If we think of a rectangle and differentiate the whole are that means I am just taking small piece of area of that rectangle. But, in physics we find a new equation by differentiating. Even, in Olympiad we differentiate some equation to find answer. When should we differentiate a equation? Does it like we are thinking smaller piece of that equation like I said at first?

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This question will likely be closed, but as a former physics olympiad contestant (and winner), I think the most common situations are:

  1. The most obvious (and trivial) situation: When we have a known function but we are interested in its derivative, we differentiate it. The majority of cases belong here.

  2. Forming and solving differential equations, such as finding from first principles the speed of transverse waves on a taut string.

  3. Obtaining an approximation of a quantity using the Taylor series (and its generalizations).

  4. Conserved quantities. For example, we have two functions of time $u(t)$ and $v(t)$, and we know that a certain combination $f(u,v) = \text{constant}$, we can differentiate it to obtain a differential equation which may be used to solve for other quantities such as $\dot{u}(t)$.

The derivative is not a small piece of something; it is the rate of change of something. On the other hand, small (infinitesimal) piece of something is more accurately known as its differential. For example, the differential of $f$ is $\text{d}f$. More accurately, $\text{d}f$ is known as the exterior derivative of $f$, which belongs to the topic of differential forms. The two concepts are closely related, however. For example, we have the standard theorem of partial derivatives $$\text{d}f = \frac{\partial f}{\partial u} \text{d}u + \frac{\partial f}{\partial v} \text{d}v$$ as well as Stokes' theorem which applies to shapes of arbitrary dimension.

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