AP Calculus AB: Essential Formulas Flashcards

Limit Definition of Derivative

• The derivative of a function f(x) at a point c is the limit of the difference quotient as h approaches 0:
  f'(c) = lim (h→0) [(f(c+h) - f(c))/h].

Alternate Definition of Derivative

• Defined as the limit:
  f'(c) = lim (x→c) [(f(x) - f(c))/(x - c)], providing an alternative perspective using the variable x instead of h.

Continuity of f(x)

• A function f(x) is continuous at a point c if:
  1. f(c) is defined,
  2. lim (x→c) f(x) exists,
  3. lim (x→c) f(x) = f(c).

Differentiability at x=c

• For f(x) to be differentiable at x=c, the left-hand and right-hand derivatives must be equal:
  f'(-)(c) = f'(+)(c).

Power Rule

• Derivative of f(x) = x^n is expressed as:
  f'(x) = nx^(n-1), where n is a constant.

Product Rule

• For two functions u(x) and v(x), the derivative is given by:
  (u·v)' = u'v + uv'.

Quotient Rule

• For functions u(x) and v(x), the derivative is:
  (u/v)' = (u'v - uv')/v^2.

Derivative of sin(x)

• Derivative of sin(x) is:
  cos(x).

Derivative of cos(x)

• Derivative of cos(x) is:
  -sin(x).

Derivative of tan(x)

• Derivative of tan(x) is:
  sec^2(x).

Derivative of cot(x)

• Derivative of cot(x) is:
  -csc^2(x).

Derivative of sec(x)

• Derivative of sec(x) is:
  sec(x)tan(x).

Derivative of csc(x)

• Derivative of csc(x) is:
  -csc(x)cot(x).

Differentiability and Continuity

• A function must be continuous to be differentiable; this means differentiability implies continuity.

Non-Differentiable Functions

• A function cannot be differentiable at points of discontinuity, corners, cusps, or vertical tangent lines.

Particle Movement Direction

• A particle moves to the left when its velocity V(t) is less than 0.

Particle at Rest

• A particle is stationary when its velocity V(t) equals 0.

Direction Change of a Particle

• A particle changes direction when the velocity V(t) changes sign, indicating crossing the x-axis on a V(t) graph.

Particle Speed Increasing

• A particle speeds up when acceleration a(t) and velocity v(t) share the same signs.

Particle Speed Decreasing

• A particle slows down when acceleration a(t) and velocity v(t) have different signs.

Chain Rule for Derivatives

• If y = f(g(x)), then the derivative is:
  dy/dx = f'(g(x))·g'(x).

Derivative of arcsin(x)

• Derivative of arcsin(x) is:
  1/√(1 - x^2).

Derivative of arccos(x)

• Derivative of arccos(x) is:
  -1/√(1 - x^2).

Derivative of arctan(x)

• Derivative of arctan(x) is:
  1/(1 + x^2).

Derivative of arccot(x)

• Derivative of arccot(x) is:
  -1/(1 + x^2).

Derivative of arcsec(x)

• Derivative of arcsec(x) is:
  1/(|x|√(x^2 - 1)).

Derivative of arccsc(x)

• Derivative of arccsc(x) is:
  -1/(|x|√(x^2 - 1)).

Derivative of ln(x)

• Derivative of ln(x) is:
  1/x.

Derivative of log base a

• Derivative of log_a(x) is:
  1/(x ln(a)).

Derivative of natural exponential (e^x)

• Derivative of e^x is:
  e^x.

Derivative of a^x

• Derivative of a^x is:
  a^x ln(a).

Derivative of Inverse Functions

• If f(b) = a and g(a) = b, then:
  g'(a) = 1 / f'(b).

Vertical Tangent Line

• A vertical tangent occurs at x=a if:
  lim (x→a) f'(x) = ±∞ and f'(x) has a nonzero numerator over zero denominator.

Vertical Asymptote

• A vertical asymptote is a line x = k where the function f(x) approaches infinity as x approaches k.

Horizontal Asymptote

• A horizontal asymptote indicates that as x approaches infinity or negative infinity, f(x) approaches a constant value.

Horizontal Tangent Line

• A horizontal tangent line indicates that the derivative f'(x) equals 0 at that point.

Increasing Function

• If f'(x) > 0, the function f(x) is increasing at that interval.

Concave Up Function

• If f''(x) > 0, the function is concave up in that interval.

Second Derivative

• The second derivative, f''(x), provides information about the concavity of the function and is used in the second derivative test for local extrema.