Differentiation: Definition and Fundamental Properties
Ten topics that turn the limit into a machine. Average rate on an interval against the instantaneous rate at a point, the difference quotient and the two limit definitions it produces, notation that moves between a value and a function, estimating a derivative from data, the four ways a derivative fails to exist, and the rules: power, constant multiple, sum and difference, then product and quotient, plus the derivatives of sine, cosine, ex and ln x, plus the four remaining trigonometric functions.
AB exam 10-15%BC exam 5-10%10 topics
Topics
Key forms For every problem in this unit
Average rate on [a, b]
(f(b) − f(a)) / (b − a), the secant slope
Derivative at a (h form)
limit of (f(a + h) − f(a)) / h as h → 0
Derivative at a (x form)
limit of (f(x) − f(a)) / (x − a) as x → a
Derivative function
f′(x): limit of (f(x + h) − f(x)) / h as h → 0
Notation, same object
f′(a) = dy/dx at x = a = d/dx[f(x)] at x = a
Symmetric estimate from data
f′(a) ≈ (f(a + h) − f(a − h)) / (2h)
Tangent line at a
y = f(a) + f′(a)(x − a)
Differentiable ⇒ continuous
the converse is FALSE: |x| at 0
No derivative at
corner, cusp, vertical tangent, discontinuity
Power
d/dx[xn] = n·xn−1, for every real n
Constant
d/dx[c] = 0, not c·x0
Constant multiple & sum
(cf)′ = c·f′; (f ± g)′ = f′ ± g′
Product
(uv)′ = u′v + uv′, NOT u′v′
Quotient
(u/v)′ = (u′v − uv′) / v2, order matters
Sine & cosine
(sin x)′ = cos x; (cos x)′ = −sin x
Exponential
(ex)′ = ex; (ax)′ = ax·ln a
Logarithm
(ln x)′ = 1/x; (logbx)′ = 1 / (x·ln b)
Tangent & cotangent
(tan x)′ = sec2x; (cot x)′ = −csc2x
Secant & cosecant
(sec x)′ = sec x tan x; (csc x)′ = −csc x cot x
Co-function pattern
every co- function carries the minus sign