Unit 2&3: Differentiation and Rate of Change CALCULUS FORMULA SHEET

Vocabulary flashcards covering differentiation and rate-of-change rules for limits, derivatives, and common functions.

Derivative (definition via limit)

The instantaneous rate of change of f at x, defined as lim_{h->0} [f(x+h) - f(x)]/h.

Derivative (limit at a)

The slope of the tangent line at x = a, defined by lim_{x->a} [f(x) - f(a)]/(x - a).

Power Rule

d/dx (x^n) = n x^(n-1).

dy/dx / f′(x) / y′ notation

The derivative of y with respect to x; often written as dy/dx, f′(x), or y′.

Second derivative

The derivative of dy/dx with respect to x: d^2y/dx^2 = f''(x).

Derivative at a (notation)

f′(a) = dy/dx |_{x=a} = y′(a).

Product rule

d/dx (u v) = u′ v + u v′.

Quotient rule

d/dx (u/v) = (v u′ - u v′) / v^2.

Chain rule

d/dx (f(g(x))) = f′(g(x)) · g′(x).

Chain rule with inner function f(y)

If y is a function of x, 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 arccos x

-1 / sqrt(1 - x^2).

d/dx arctan x

1 / (1 + x^2).

d/dx arccsc x

-1 / (|x| sqrt(x^2 - 1)).

d/dx arcsec x

1 / (|x| sqrt(x^2 - 1)).

d/dx arccot x

-1 / (1 + x^2).

Mnemonic: d/dx (Success) = Effort

A personal mnemonic from the notes; not a standard calculus rule.