AP Calculus AB Unit 2 Notes: Learning the Core Differentiation Toolkit

Power Rule

If f(x)=x^n for any real n, then f'(x)=n·x^(n−1).

Power Function

A function built from terms of the form x^n (n can be positive, negative, or fractional).

Exponent (n) in the Power Rule

The number n in x^n; it becomes the new coefficient in the derivative and the exponent decreases by 1.

Instantaneous Rate of Change

The derivative at a point; the slope of the tangent line to the curve at that input.

Limit Definition of the Derivative

f'(x)= ext{lim}_{h o0} \frac{f(x+h)−f(x)}{h}; measures change in f over a tiny change in x.

Constant Rule

d/dx(c)=0 for any constant c; a constant function has slope 0.

Constant Multiple Rule

d/dx(c·f(x))=c·f'(x); constants factor out of derivatives.

Sum Rule

d/dx(f(x)+g(x))=f'(x)+g'(x); differentiate term-by-term across addition.

Difference Rule

d/dx(f(x)−g(x))=f'(x)−g'(x); negatives must be tracked carefully.

Linearity (of derivatives)

The collection of rules showing derivatives distribute over addition/subtraction and pull out constant multiples (sum/difference/constant multiple rules).

Derivative Notation: f'(x)

The derivative of a function named f, written as a function of x.

Derivative Notation: dy/dx

Derivative of y with respect to x; common when given an equation like y = (expression).

Derivative at a Point: f'(a)

The value of the derivative when x=a; equals the slope of the tangent line at x=a.

Derivative of sin x

d/dx( ext{sin} x)= ext{cos} x (assuming x is in radians).

Derivative of cos x

d/dx( ext{cos} x)=− ext{sin} x (a common sign-error point).

Derivative of tan x

d/dx( an x)=\frac{1}{ ext{cos}^2 x}.

Derivative of sec x

d/dx(sec x)=sec x·tan x.

Derivative of csc x

d/dx(csc x)=−csc x·cot x.

Derivative of cot x

d/dx( ext{cot} x)=−\frac{1}{ ext{sin}^2 x}.

Radian Measure Requirement (Trig Derivatives)

Standard trig derivative formulas are valid when angles are measured in radians; degrees would introduce conversion factors.

Trig Domain Restrictions (tan, sec)

tan x and sec x are undefined where cos x = 0, and their derivatives share those restrictions.

Trig Domain Restrictions (csc, cot)

csc x and cot x are undefined where sin x = 0, and their derivatives share those restrictions.

Derivative of e^x

d/dx(e^x)=e^x; base e is special because the function equals its own derivative.

Derivative of a^x (a>0, a≠1)

d/dx(a^x)=a^x ext{ln}(a); includes the important ext{ln}(a) factor.

Derivative of ln x

d/dx( ext{ln} x)=\frac{1}{x}; valid only for x>0 in the real-number setting.