The Core Concept: It's a Rate of Change

At its heart, the derivative of A with respect to B measures how much A changes when B changes by a very small amount.

In the classic derivative you learn first, dy/dx, you are asking: "As I change the independent variable x by a tiny amount, how much does the dependent variable y change?"

The phrase "with respect to" tells you which variable is playing the role of the independent variable (the one you are changing, the input) and which is the dependent variable (the one that changes as a result, the output).

  • A is the dependent variable (the "output" or function's value).
  • B is the independent variable (the "input" or the variable causing the change).

So, dA/dB answers the question: "If I nudge B a little, what is the ratio of the resulting nudge in A?"

The Formal, Mathematical Definition

Let's formalize this. If A is a function of B, we can write A = f(B).

The derivative of A with respect to B, denoted as dA/dB or f'(B), is defined by the limit:

dA/dB = lim (ΔB → 0) [ f(B + ΔB) - f(B) ] / ΔB

Where: * ΔB (pronounced "delta B") represents a very small change in the variable B. * f(B + ΔB) - f(B) represents the corresponding change in the variable A (which is ΔA). * The entire fraction ΔA / ΔB is the average rate of change over a small interval. * The limit as ΔB → 0 "zooms in" on an infinitesimally small change, giving us the instantaneous rate of change.

This limit is the slope of the tangent line to the curve A = f(B) at a specific point.

A Concrete Example: Geometry

Let's say A represents the area of a square and s represents the length of its side. We know the formula: A = s²

Here, Area (A) depends on the side length (s). So, let's find the derivative of A with respect to s, which is dA/ds.

  1. Identify the Function: A = f(s) = s²
  2. Apply the Definition: We want dA/ds = f'(s). Using the power rule (a shortcut from the limit definition), the derivative is: dA/ds = 2s

What does this mean physically?

The derivative dA/ds = 2s tells us the rate at which the area of a square changes as its side length changes.

For a specific square with a side length of 5 units: * dA/ds = 2 * 5 = 10 * Interpretation: If you were to increase the side length s from 5 units, the area A would be increasing at a rate of 10 square units for every one unit increase in side length at that exact moment.

It's like the "sensitivity" of the area to a small change in the side length.

This concept becomes incredibly powerful in "related rates" problems, where multiple variables change over time. Let's use the standard variables x and y.

Imagine a ladder sliding down a wall. * Let x be the distance from the base of the ladder to the wall. * Let y be the height of the top of the ladder on the wall. * The length of the ladder is fixed, say 10 ft. So, by the Pythagorean Theorem: x² + y² = 10²

Now, suppose the base is pulled away at 2 ft/s. This is dx/dt = 2 (the derivative of x with respect to time).

We can ask: What is dy/dt (the derivative of y with respect to time) when x = 6 ft?

  1. We have a relationship: x² + y² = 100

  2. Differentiate both sides with respect to time (t). This is called implicit differentiation. d/dt (x²) + d/dt (y²) = d/dt (100) 2x * (dx/dt) + 2y * (dy/dt) = 0

  3. Plug in what we know:

    • x = 6 ft
    • dx/dt = 2 ft/s
    • Find y when x=6: 6² + y² = 100 => y² = 64 => y = 8 ft (we take the positive root since it's a height)

  4. Solve for dy/dt: 2*(6)*(2) + 2*(8)*(dy/dt) = 0 24 + 16(dy/dt) = 0 16(dy/dt) = -24 dy/dt = -1.5 ft/s

Interpretation: The derivative of y with respect to time is -1.5. This means the top of the ladder is sliding down the wall at a rate of 1.5 feet per second at the exact instant when the base is 6 feet from the wall.

Summary and Key Takeaways

  1. Meaning: The derivative of A with respect to B (dA/dB) is the instantaneous rate of change of A relative to a change in B.
  2. Graphical Interpretation: It represents the slope of the tangent line to the curve A = f(B).
  3. Notation: It can be written as dA/dB, f'(B), or A'(B).
  4. "With Respect To": This phrase is crucial. The derivative of A = s² with respect to s is 2s, but its derivative with respect to time (t) would be 2s * (ds/dt), which is a completely different quantity.
  5. Universality: This concept applies to any two related quantities—distance over time, cost over units produced, energy over temperature, etc. It is the fundamental tool for understanding how interconnected systems behave.