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37 Cards in this Set

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definition of the derivative
F'(x) = lim as h approaches 0
[f(x+h) - f(x)] / h
derivative of a constant times a function
d/dx (k(u(x))) = k du/dx
Power Rule (variable raised to a constant)
d/dx (u^n) = nu^(n-1) du/dx
The Sum Rule
d/dx (u+v) = du/dx + dv/dx
The Difference Rule
d/dx (u-v) = du/dx - dv/dx
The Product Rule
d/dx (uv) = uv' + vu'
The Quotient Rule
d/dx (u/v) = [vu' - uv'] / v^2
The Chain Rule
dy/dx = dy/du * du/dx
or
d/dx (f(g(x)) = f'(g(x))g'(x)
The Derivative of the Sine
d/dx (sinu) = cosu du/dx
The Derivative of the Cosine
d/dx (cosu) = -sinu du/dx
The Derivative of the Tangent
d/dx (tanu) = sec^2 u du/dx
The Derivative of the Cotangent
d/dx (cotu) = -csc^2 u
The Derivative of the Secant
d/dx (secu) = secu tanu du/dx
The Derivative of the Cosecant
d/dx (cscu) = -cscu cotu du/dx
The Derivative of the Inverse Sine
d/dx (Sin^-1 u) = 1/sqrt(1-u^2) du/dx
The Derivative of the Inverse Cosine
d/dx(Cos^-1 u) = -1/sqrt(1-u^2) du/dx
The Derivative of the Inverse Tangent
d/dx(Tan^-1 u) = 1/(1+u^2) du/dx
The Derivative of the Inverse Cotangent
d/dx(Cot^-1 u) = -1/(1+u^2) du/dx
The Derivative of the Inverse Secant
d/dx(Sec^-1 u) = 1/(|u|sqrt(u^2 - 1) du/dx
The Derivative of the Inverse Cosecant
d/dx(Csc^-1 u) = -1/(|u|sqrt(u^2 - 1) du/dx
The Derivative of the Natural Log
d/dx (ln u) = 1/u du/dx
The Derivative of the log to base a
d/dx (loga u) = 1/(u lna) du/dx
The Derivative of e raised to a variable
d/dx (e^u) = e^u du/dx
The Derivative of a constant raised to a variable
d/dx (a^u) = a^u lna du/dx
Pythagorean Identities
sin^2 x + cos^2 x = 1
tan^2 x +1 = sec^2 x
1 + cot^2 x = csc^2 x
sin even/odd formula
sin (-x) = -sinx
cos even/odd formula
cos(-x) = cosx
tan even/odd formula
tan(-x) = -tanx
csc even/odd formula
csc(-x) = -cscx
sec even/odd formula
sec(-x) = secx
cot even/odd formula
cot(-x) = -cotx
double angle formula: sin(2x)
= 2sinx cosx
double angle formula: cos(2x)
= cos^2 x - sin^2 x
= 2cos^2 x - 1
= 1-2sin^2 x
half angle formula: cos^2 x
= (1/2)(1+cos(2x))
double angle formula: tan(2x)
= (2tanx)/(1-tan^2 x)
half angle formula: sin^2 x
= (1/2)(1-cos(2x))
half angle formula: tan^2 x
= (1-cos(2x))/(1+cos(2x))