Unit 2: Differentiation — Definition & Fundamental Properties

Power, Product & Quotient Rules

Differentiating using the power, product, and quotient rules.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Defining the Derivative

Three Rules for Differentiating Quickly

Computing f'(x) = 2x + 3 from the limit definition took several steps. Every one of these faster rules can be derived from that same definition — they aren't separate assumptions.

Definition — Power, Product, and Quotient Rules

d/dx[xⁿ] = n·xⁿ⁻¹ (from expanding (x+h)ⁿ and letting h → 0, every cross term but nxⁿ⁻¹h vanishing). (fg)' = f'g + fg' (from an add-and-subtract trick on the product's difference quotient). (f/g)' = (f'g − fg')/g² — derivable from the product rule itself, by writing h = f/g so f = hg, differentiating both sides, and solving for h'.

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Worked Example — Power and Sum Rules

Differentiate f(x) = 5x⁴ − 2x³ + x − 7. Term by term: f'(x) = 20x³ − 6x² + 1 (the constant −7 contributes 0).

Worked Example — Product Rule

Differentiate f(x) = (2x − 1)(x² + 3). With u = 2x − 1, v = x² + 3: u' = 2, v' = 2x. f' = u'v + uv' = 2(x² + 3) + (2x − 1)(2x) = 2x² + 6 + 4x² − 2x = 6x² − 2x + 6. Check by expanding first: (2x−1)(x²+3) = 2x³ − x² + 6x − 3, and differentiating directly gives 6x² − 2x + 6 — matching.

Worked Example — Quotient Rule

Differentiate g(x) = (x² + 1)/(2x − 3). With u = x² + 1, v = 2x − 3: u' = 2x, v' = 2. g' = (u'v − uv')/v² = [2x(2x−3) − (x²+1)(2)] / (2x−3)² = [4x² − 6x − 2x² − 2] / (2x−3)² = (2x² − 6x − 2)/(2x−3)².

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

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

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
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

Verify a product or quotient rule result by expanding or rewriting the original expression where possible — two independent methods agreeing is strong evidence neither has an error.

Common Mistakes

Differentiating a product as f'(x)g'(x), multiplying the two derivatives together.
The product rule has two full terms, f'g + fg' — there's no shortcut that multiplies the derivatives alone.
Reversing the subtraction order in the quotient rule numerator.
It's f'g − fg', derivative-of-numerator first — reversing it flips the sign of the whole result.

Key Takeaways

The power, product, and quotient rules all trace back to the limit definition of the derivative.
The quotient rule can itself be derived from the product rule.
Expanding a product or rewriting a quotient, where possible, is a reliable way to check a result.

Summary

This closes Unit 2. Unit 3 extends differentiation to functions composed inside other functions, and to relations that aren't solved for y.

Track your progress

← Defining the Derivative Back to Unit 2: Differentiation — Definition & Fundamental Properties