Derivative as a Limit: The Difference Quotient Explained Step-by-Step

In calculus, the derivative measures the instantaneous rate of change of a function with respect to its variable. While algebra allows us to calculate the average rate of change between two distinct points on a curve, calculus allows us to calculate the exact slope at a single instantaneous point.

The derivative of a function f(x) at any point x is defined as the limit of the difference quotient as h (or Δx) approaches zero:

f'(x) = lim (h -> 0) [ (f(x + h) - f(x)) / h ]

Geometric Interpretation:

• The ratio [f(x + h) - f(x)] / h represents the slope of the secant line connecting point (x, f(x)) and point (x+h, f(x+h)).
• As h → 0, the second point slides along the curve toward the first point, and the secant line morphs smoothly into the tangent line.

1. Write the Definition:

   f'(x) = lim (h -> 0) [ ((x + h)² - x²) / h ]

2. Expand the Binomial:

   (x + h)² = x² + 2xh + h²

3. Substitute and Simplify Numerator:

   [ (x² + 2xh + h²) - x² ] / h = (2xh + h²) / h

4. Factor out h:

   h(2x + h) / h = 2x + h   (for h ≠ 0)

5. Evaluate the Limit as h → 0:

   lim (h -> 0) (2x + h) = 2x + 0 = 2x

Thus, we have proven from first principles that the derivative of f(x) = x² is f'(x) = 2x (the foundational Power Rule).

• Dividing by zero before simplifying: Never set h = 0 in the denominator immediately; always factor and cancel h first.
• Forgetting parenthesis in f(x + h): For f(x) = 3x - 5, write 3(x + h) - 5, not 3x + h - 5.