Formula Quiz #2 (key)

Constant Rule

The derivative of a constant c is 0.

Power Rule

The derivative of x^n is n*x^(n-1).

Constant Multiple Rule

The derivative of cx^n is cn*x^(n-1).

Sum & Difference Rule

The derivative of u(x) + v(x) is u'(x) + v'(x).

Chain Rule

The derivative of f(g(x)) is f'(g(x)) * g'(x).

Product Rule

The derivative of f(x)*g(x) is f(x)*g'(x) + g(x)*f'(x).

Quotient Rule

The derivative of f(x)/g(x) is [g(x)*f'(x) - f(x)*g'(x)] / [g(x)]^2.

Derivative of sin(x)

The derivative of sin(x) is cos(x).

Derivative of cos(x)

The derivative of cos(x) is -sin(x).

Derivative of tan(x)

The derivative of tan(x) is sec^2(x).

Derivative of sec(x)

The derivative of sec(x) is sec(x)tan(x).

Derivative of csc(x)

The derivative of csc(x) is -csc(x)cot(x).

Derivative of cot(x)

The derivative of cot(x) is -csc^2(x).

Derivative of e^x

The derivative of e^x is e^x.

Derivative of a^x

The derivative of a^x is a^x * ln(a).

Derivative of ln(x)

The derivative of ln(x) is 1/x.

Intermediate Value Theorem

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c in (a, b) such that f(c) = k.

Continuous Function

A function f is continuous at x = a if f(a) exists, limits exist at f(x), and f(a) = limit as x approaches a of f(x).

Limit Definition of a Derivative

The derivative of f at x is the limit as h approaches 0 of [f(x + h) - f(x)] / h.

Average Rate of Change

The average rate of change of f(x) on [a, b] is [f(b) - f(a)] / [b - a].

Double-Angle Identity (sin(2x))

= 2sin(x)cos(x).

Double-Angle Identity (cos(2x))

= cos^2(x) - sin^2(x) or 2cos^2(x) - 1 or 1 - 2sin^2(x).

Pythagorean Identity

sin^2(x) + cos^2(x) = 1.

Area of Circle

A = πr^2.

Circumference of Circle

C = 2πr.

Volume of Prism

V = Bh, where B = area of the base.

Volume of Right Circular Cone

V = (1/3)πr^2h.

Volume of Sphere

V = (4/3)πr^3.

Volume of Right Circular Cylinder

V = πr^2h.

Surface Area of Cylinder

SA = 2πrh + 2πr^2.

Volume of Cube

V = s^3.

Area of Non-Right Triangle

A = (1/2)ab sin(C).

Rolle's Theorem

If f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) such that f'(c) = 0.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = [f(b) - f(a)] / [b - a].

Extreme Value Theorem

If f is continuous on [a, b], then f has an absolute maximum and minimum on [a, b].

Derivative of log_bx

d/dx(log_bx) = 1/xlnb

Derivative f^-1(x)

d/dx(f^-1(x)) = 1/f’(f^-1(x))

d/dx arcsinx

= 1/√(1-x²)

d/dx arccosx

= -1/√(1-x²)

d/dx arctanx

= 1/(1+x²)

d/dx arccscx

= -1/(|x|√(x²-1))

d/dx arcsecx

= 1/(|x|√(x²-1))

d/dx arccotx

= -1/(1+x²)

double angle identity sin²x

= 1-cos(2x)/2

double angle identity cos²x

= 1+cos(2x)/2

alternative form of derivative at x=c

the limit of x as it approaches c is f(x)-f(c)/x-c