Unit 2&3: Differentiation and Rate of Change - Vocabulary Flashcards

Vocabulary flashcards covering the key differentiation formulas and notation from the calculus formula sheet.

Derivative

The instantaneous rate of change of a function; defined as lim_{h→0} (f(x+h)−f(x))/h.

Derivative at a point

Slope of the tangent at x=a: dy/dx|{x=a} = lim{x→a} (f(x)−f(a))/(x−a).

Power rule

d/dx (x^n) = n x^{n−1}.

Derivative notation

dy/dx, f′(x), and ẏ all denote the derivative of y with respect to x.

Second derivative

d^2y/dx^2 = f''(x) = ÿ; the derivative of the derivative.

Derivative value at a point

f′(a) = dy/dx|_{x=a}; the slope of f at x=a.

Product rule

d/dx (u·v) = u′v + uv′.

Quotient rule

d/dx (u/v) = (v u′ − u v′) / v^2.

Chain rule (composition)

d/dx (f(g(x))) = f′(g(x)) · g′(x).

Chain rule (inner function) alternate

d/dx (f(y)) = f′(y) · dy/dx.

Derivative of inverse function

(f^−1)′(x) = 1 / f′(f^−1(x)).

d/dx (sin x)

cos x.

d/dx (cos x)

−sin x.

d/dx (tan x)

sec^2 x.

d/dx (csc x)

−csc x cot x.

d/dx (sec x)

sec x tan x.

d/dx (cot x)

−csc^2 x.

d/dx (e^x)

e^x.

d/dx (a^x)

a^x ln a.

d/dx (ln x)

1/x.

d/dx (log_a x)

1 / (x ln a).

d/dx (arcsin x)

1 / sqrt(1 − x^2).

d/dx (arctan x)

1 / (1 + x^2).

Limit definition at a point (x→a)

lim_{x→a} [f(x)−f(a)]/(x−a) = f′(a).