The First Principles technique focuses on decomposing a question to its fundamentals. This is essential for calculus and vectors.

Example

A spherical balloon is being inflated. Information is given that the surface area $A$ is increasing at a constant rate of $24\text{ cm}^2/\text{s}$. We need to find the rate at which the radius $r$ is increasing at the specific moment when the radius is $3\text{ cm}$.

To work on this with First Principles, we want to:

• Identify what we know (expressed as a derivative, e.g., $d?/dt$).
• Identify what we want to find (expressed as a derivative).
• Find a “bridge” equation that connects the variables involved.

Using the question - we know that:

• The surface area is increasing at $24\text{ cm}^2/\text{s}$
• $A$ is area
• $t$ is time
• $r$ is radius, specifically measured at 3cm (can be treated as a constant)

Therefore, we need to identify “rate of change of area” as a derivative.

See anx Equation