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Daily Math Minute

Derivatives

Power, Product & Quotient Rules

Differentiating using the power, product, and quotient rules.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Interpreting the Derivative

Faster Rules, Built on the Same Definition

Computing f'(x) = 2x − 4 from the limit definition took several steps. Before reading on: the pattern '2 times x to the power one less than 2' suggests a shortcut — does it generalize to any power of x?

It does, and it isn't a coincidence. Expanding (x+h)ⁿ, every term carries at least one factor of h except the leading n·xⁿ⁻¹h term — so dividing the difference quotient by h and letting h → 0 always leaves n·xⁿ⁻¹.

Definition — Power, Product, and Quotient Rules

d/dx[xⁿ] = n·xⁿ⁻¹. Constants factor out, and derivatives distribute over sums. For products: (fg)' = f'g + fg'. For quotients: (f/g)' = (f'g − fg')/g².
(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Worked Example — Power and Sum Rules

Differentiate f(x) = 2x⁵ − 6x³ + 4x − 9. Term by term: f'(x) = 10x⁴ − 18x² + 4.

Worked Example — Product Rule

Differentiate g(x) = (3x − 2)(x² + 4). With u = 3x − 2, v = x² + 4: u' = 3, v' = 2x. g' = u'v + uv' = 3(x²+4) + (3x−2)(2x) = 3x² + 12 + 6x² − 4x = 9x² − 4x + 12. Check by expanding first: (3x−2)(x²+4) = 3x³ − 2x² + 12x − 8, and differentiating directly gives 9x² − 4x + 12 — matching.

Worked Example — Quotient Rule

Differentiate h(x) = (4x + 1)/(x − 2). With u = 4x+1, v = x−2: u' = 4, v' = 1. h' = (u'v − uv')/v² = [4(x−2) − (4x+1)(1)]/(x−2)² = (4x − 8 − 4x − 1)/(x−2)² = −9/(x−2)².

Derivative Explorer

Point of tangency
1

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
  1. At x = 1, f(x) ≈ -2 — the point (1, -2).
  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.
  3. That slope is positive, so f is increasing at this point.
  4. The second derivative is positive here, so f is concave up (curving upward) near this point.
  5. 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.
hForwardBackwardCentral
14-21
0.10.31-0.290.01
0.010.0301-0.02990.0001
0.0010.003-0.0030
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

Check a product-rule result by expanding first and differentiating directly, when the expression is simple enough — two methods landing on the same answer 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 multiplying the derivatives alone.

  • Reversing the subtraction order in the quotient rule's numerator.

    It's f'g − fg', derivative-of-numerator first — reversing it flips the sign of the entire result.

Key Takeaways

  • d/dx[xⁿ] = n·xⁿ⁻¹ follows from the binomial expansion used in the difference quotient.
  • The product rule is f'g + fg'; the quotient rule is (f'g − fg')/g² — neither is simply the product or quotient of the two derivatives.

Summary

These rules handle sums, products, and quotients of power terms. The next lesson handles a function wrapped inside another.

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