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Level 420

Derivative Rules


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The derivative of ƒ
ƒ'(x) = lim ƒ(x + h) − ƒ(x) / h h→0
ƒ'(a)
tells the slope of (a, ƒ(a))
derviative of a constant: f(x) = 5?
the limit of a constant is always 0
f'(x) = 0
f(x) = c (derivative of a constant function)
f'(x) = nxⁿ⁻¹
f(x) = xⁿ (the power rule)
f'(x) = Cnxⁿ⁻¹
f(x) = Cxⁿ (the constant multipler rule)
f'(x) ± g'(x)
(f ± g)(x) (the sum & difference rules)
x→∞
the number, e
y=e^x
y'=e^x
h'(x) = f(x)g'(x) + g(x)f'(x)
h(x) = f(x) * g(x) (product rule)
h(x) = g(x) / f(x) (quotient rule)
h'(x) = g(x)f'(x) - f(x)g'(x) / g²(x)
∆s(t)/∆t
average velocity (aka average speed)
instantaneous velocity
lim ∆s(t) / ∆t
s'(t)
position of a new location
∆v(t)/∆t
average acceleration
instantaneous acceleration
lim ∆v(t) / ∆t
definition of the number e
lim (e^h - 1) / h = 1
f(x) = sin x
f'(x) = cos x
squeeze theorem of sin
lim sin θ / θ = 1
squeeze theorem of cos
lim (cos θ -1) / θ = 0
y = cos x
y' = -sin x
y' = sec²x
y = tan x (= sin x / cos x)
y' = sec x tan x
y = sec x (= 1 / cos x)
y = csc x
y' = -csc x cot x
y' = -csc²x
y = cot x
y = sin⁻¹ x
y' = 1 / √(1 - x²)
y = cos⁻¹ x
y' = -1 / √(1 - x²)
y = tan⁻¹x
y' = 1 / 1 + x²
y = csc⁻¹x
y' = - 1 / x√(x² - 1)
y = sec⁻¹x
y' = 1 / x√(x² - 1)
y = cot⁻¹x
y' = - 1 / 1 + x²
csc x
1 / sin x
tan x
sin x / cos x
cot x
1 / tan x
sec x
1 / cos x
cot x
cos x / sin x
1
sin²x + cos²x
sec²x
1 + tan²x
csc²x
1 + cot²x
sin 2x
2 sin x cos x
cos 2x
cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x
tan 2x
2 tan x / 1- tan²x
sin²x
(1 - cos 2x) / 2
cos²x
(1 + cos 2x) / 2