Understanding Derivatives: From Slope of a Curve to Instantaneous Velocity

What Is a Derivative?

• The derivative is the slope of a curve at a single point. It tells you how fast a function is changing at that exact location.
• In calculus we write the derivative of a function (f(x)) as (f'(x)), (\frac{df}{dx}), or simply (F'(x)).

Limit Definition

The formal definition uses the difference quotient: [f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}] - As (h) approaches zero, the two points on the graph get infinitely close, turning the secant line into the tangent line. - Dropping the specific (x) value (i.e., not plugging in a number) gives a derivative function that works for any (x).

Computing Derivatives – Step‑by‑Step

1. Identify (f(x)).
2. Form (f(x+h)) by replacing every (x) with (x+h).
3. Subtract (f(x)) and place the result over (h).
4. Expand, combine like terms, and factor out (h).
5. Cancel (h) and take the limit as (h\to0).

Example 1: Polynomial

For (f(x)=2x^2-3): - (f(x+h)=2(x+h)^2-3) - After algebra, the limit simplifies to (f'(x)=4x). - The slope at (x=2) is (4\times2=8).

Example 2: Cubic Function

For (f(x)=2x^3-x): - Expand ((x+h)^3) using the binomial theorem. - After canceling (h), the derivative is (f'(x)=6x^2-1).

Example 3: Square‑Root Function

For (f(x)=\sqrt{x}): - Multiply numerator and denominator by the conjugate to eliminate the radical. - The derivative becomes (f'(x)=\frac{1}{2\sqrt{x}}).

Tangent Line Equation

Once you have (f'(a)) (the slope at (x=a)) and the point ((a, f(a))), use the point‑slope form: [y - f(a) = f'(a)(x - a)] This yields a straight line that touches the curve only at ((a, f(a))).

Applications

• Instantaneous Velocity: The derivative of a position function (s(t)) gives velocity (v(t)=s'(t)).
• Example: (s(t)=1250-16t^2) → (v(t)=-32t). Setting (s(t)=0) finds the time of impact; plugging that time into (v(t)) gives the impact speed.

Differentiability vs. Continuity

• A function must be continuous to be differentiable, but continuity alone does not guarantee differentiability.
• Sharp corners (e.g., (|x|) at (x=0)) and vertical tangents cause the limit of the slope to fail, making the derivative undefined.
• If the left‑hand and right‑hand limits of the difference quotient are unequal, the derivative does not exist at that point.

Notation Cheat Sheet

Key Steps for Any Function

1. Write the difference quotient.
2. Expand and simplify algebraically.
3. Factor out (h) and cancel.
4. Take the limit as (h\to0).
5. Use the resulting derivative function for slopes, tangent lines, or rates of change.

Why It Matters

Understanding the derivative bridges geometry (slopes), physics (velocity, acceleration), and real‑world modeling (growth rates, optimization). Mastery of the limit process prevents mechanical computation errors and deepens conceptual insight.

A derivative is simply the instantaneous slope of a function at a point; mastering its limit definition lets you compute slopes, tangent lines, and real‑world rates of change for any differentiable function.