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UNIT 2 · 4–7%

Definition & Basic Rules

turning "slope of a secant" into "slope of a tangent" 📉

📖 derivative definitions

f′(x) = limh→0 [f(x+h) − f(x)] / h
Alternate: f′(a) = limx→a [f(x) − f(a)] / (x − a)

Geometric meaning: f′(a) is the slope of the tangent line to y = f(x) at x = a.

⚡ power rule & constants

d/dx [xⁿ] = n xⁿ⁻¹   (n any real number in BC where defined)

Constant multiple: (cf)′ = c f′. Sum rule: (f + g)′ = f′ + g′.

✖️➗ product & quotient rules

(fg)′ = f′g + f g′
(f/g)′ = (f′g − f g′) / g²

📐 trig derivatives (memorize!)

  • d/dx [sin x] = cos x  ·  d/dx [cos x] = −sin x
  • d/dx [tan x] = sec²x  ·  d/dx [sec x] = sec x tan x
  • d/dx [cot x] = −csc²x  ·  d/dx [csc x] = −csc x cot x

🔗 exponentials & logs

d/dx [eˣ] = eˣ   ·   d/dx [ln x] = 1/x (x > 0)

For ln|x|, derivative is still 1/x on each interval where defined.

✏️ quick practice

  1. Differentiate f(x) = 3x⁴ − 2x + 7.
  2. Find y′ if y = x² sin x (product rule).
  3. Use the limit definition of derivative for f(x) = 2x + 1 at x = 3 (spoiler: should get 2).
← unit 1 home unit 3 →

chain rule comes next… get ready 🔗