Unit 2: Differentiation — Definition & Fundamental Properties
Power & Sum Rules
Differentiating using the power rule and sum/difference rules.
Prerequisites
- Differentiability & Continuity
A Shortcut, Derived from the Definition
Computing f'(x) = 2x for f(x) = x² from the limit definition took several algebra steps. Before reading on: is there a pattern in that result — 2x from x², matching '2 times x to the power one less than 2' — that might generalize to any power of x?
The pattern does generalize, and it isn't a coincidence — it falls straight out of the same difference-quotient process. For f(x) = x³, expand (x + h)³ = x³ + 3x²h + 3xh² + h³ using the binomial expansion. Then [f(x+h) − f(x)]/h = (3x²h + 3xh² + h³)/h = 3x² + 3xh + h², and as h → 0, every remaining term with a factor of h vanishes, leaving exactly 3x² — matching '3 times x to the power one less than 3.' The same binomial-expansion argument works for any positive integer power n: every cross term in (x + h)ⁿ carries at least one factor of h except the leading nxⁿ⁻¹h term, so dividing by h and letting h → 0 always leaves n·xⁿ⁻¹.
Definition — Power, Constant Multiple, and Sum/Difference Rules
Worked Example — Differentiating a Polynomial
Worked Example — Rewriting Radicals and Fractions as Powers First
Derivative Explorer
Point of tangency
f(x), with tangent line at x = a
Derivative graph — linked to the graph above; panning or zooming either moves both
What's happening at x = a
- At x = 1, f(x) ≈ -2 — the point (1, -2).
- The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
- That slope is positive, so f is increasing at this point.
- The second derivative is positive here, so f is concave up (curving upward) near this point.
- Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.
Numerical derivative comparison
| h | Forward | Backward | Central |
|---|---|---|---|
| 1 | 4 | -2 | 1 |
| 0.1 | 0.31 | -0.29 | 0.01 |
| 0.01 | 0.0301 | -0.0299 | 0.0001 |
| 0.001 | 0.003 | -0.003 | 0 |
Analysis of f(x)
- y-intercept
- (0, 0)
- x-intercepts
- (-1.73, 0), (0, 0), (1.73, 0)
- Extrema
- local max at (-1, 2); local min at (1, -2)
- Inflection points
- (0, 0)
- Vertical asymptotes
- none found in view
- Horizontal asymptotes
- none found
- Domain
- all real numbers in view
- Range (estimated)
- approximately [-970, 970]
Analysis of f'(x) — its roots are f's critical points
- y-intercept
- (0, -3)
- x-intercepts
- (-1, 0), (1, 0)
- Extrema
- local min at (0, -3)
- Inflection points
- none found in view
- Vertical asymptotes
- none found in view
- Horizontal asymptotes
- none found
- Domain
- all real numbers in view
- Range (estimated)
- approximately [-3, 297]
Tip
Common Mistakes
Applying the power rule to a constant term and getting a nonzero result.
A constant c can be written as c·x⁰, and the power rule gives c·0·x⁻¹ = 0 — a constant's derivative is always 0, since its rate of change is always 0.
Forgetting to reduce the exponent by 1 in the new coefficient, writing d/dx[5x⁻²] as −10x⁻² instead of −10x⁻³.
The power rule reduces the exponent by exactly 1 regardless of its sign: n = −2 becomes n − 1 = −3, so the correct derivative is 5 · (−2)x^(−2−1) = −10x⁻³.
Key Takeaways
- d/dx[xⁿ] = n·xⁿ⁻¹ follows directly from the limit definition, via the binomial expansion of (x+h)ⁿ.
- Constants factor out of derivatives, and derivatives distribute over sums and differences — a polynomial can be differentiated term by term.
- Rewriting radicals and fractions as powers of x (fractional or negative exponents) lets the power rule handle them directly.
Summary
The power rule handles sums of power terms quickly, but not products or quotients of two whole functions. The next lesson derives rules for exactly that.
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