AP Calculus AB: Formulas, Theorems, and Procedures for Independent Study

Derivative existence conditions

The graph must be smooth, not discontinuous, no corner, no cusp, and no vertical tangent.

Average Rate of Change formula

a.r.c. = (f(b) - f(a)) / (b - a)

Definition of the Derivative

f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]

Instantaneous Rate of Change

The slope of the tangent line at a point.

Notation for Instantaneous Rate of Change

f'(a) = dy/dx |_(x=a)

Derivative of a constant function

d/dx[c] = 0 for any constant c.

Derivative of a linear function

d/dx[ax + b] = a for any constants a and b.

Power Rule for derivatives

d/dx[x^n] = n*x^(n-1) for any real number n.

Derivative of cx^n

d/dx[cx^n] = cnx^(n-1) for any constant c.

Derivative of sin(x)

d/dx[sin(x)] = cos(x).

Derivative of cos(x)

d/dx[cos(x)] = -sin(x).

Derivative of tan(x)

d/dx[tan(x)] = sec^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[cot(x)] = -csc^2(x).

Derivative of e^x

d/dx[e^x] = e^x.

Derivative of ln(x)

d/dx[ln(x)] = 1/x.

Derivative of kx

d/dx[kx] = k for any constant k.

Writing a tangent line

You need a point and a slope: y = f(a) + f'(a)(x - a).

Writing a normal line

You need a point and a perpendicular slope: y = f(a) - (1/f'(a))(x - a).

Finding a Linear Approximation

Write a tangent line and plug in your x-value of your approximation to find y.

Justification for horizontal tangent lines

f(x) has horizontal tangents when dy/dx = 0.

Justification for vertical tangent lines

f(x) has vertical tangents when dy/dx is undefined.

Justification for linear approximation estimates

A linear approximation is an overestimate if the curve is concave down, and an underestimate if the curve is concave up.

Derivative of Inverse Function

The derivative of an inverse function is the reciprocal of the derivative of the original function at the 'matching' point.

Chain Rule

If dy/du = dy/dx dx/du, then dy/dx = dy/du du/dx.

Product Rule

d(f(x)g(x))/dx = f(x)g'(x) + g(x)f'(x).

Derivatives of Inverse Trig Functions

d/dx(sin^-1(x)) = 1/sqrt(1-x^2), d/dx(cos^-1(x)) = -1/sqrt(1-x^2), d/dx(tan^-1(x)) = 1/(1+x^2).

Derivatives of Exponential Functions

d/dx(a^x) = a^x ln(a).

Derivatives of Logarithmic Functions

d/dx(log_a(x)) = 1/(x ln(a)).

Derivatives of Generic Functions

d/dx(f(x)) = f'(x).

Properties of Derivatives

d(cf(x))/dx = c * d(f(x))/dx.

Derivatives of Inverse Functions

If (a, b) is on f(x), then (b, a) is on f^-1(x).