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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 logbx
d/dx(logbx) = 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
