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17 Cards in this Set

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lim(x→a) [f(x) + g(x)] =
lim(x→a)f(x) + lim(x→a)g(x)
lim(x→a) [c f(x)] =
c lim(x→a) f(x)
lim(x→a) [f(x) ÷ g(x)] =
lim(x→a)f(x) ÷ lim(x→a)g(x)

if lim(x→a)g(x) ≠ 0
lim(x→a) [f(x)]n =
[ lim(x→a) f(x) ]^n

where n is a positive integer
The Squeeze Theorem
If f(x) ≤ g(x) ≤ h(x)
when x is near a (except possibly at a) and
lim(x→a) f(x) = lim(x→a) h(x) = L
then lim(x→a) h(x) = L
(derivative of a constant function)

d/dx (c) =
0
d/dx (x) =
1
(power rule)

d/dx (x^n) =
nx^(n-1)
(the constant multiple rule)

d/dx [cf(x)] =
c d/dx f(x)
(the sum rule)

d/dx [f(x) + g(x)] =
d/dx f(x) + d/dx g(x)
(the difference rule)

d/dx [f(x) - g(x)] =
d/dx f(x) - d/dx g(x)
(the product rule)

d/dx [f(x)g(x)] =
f(x) d/dx[ g(x)] + g(x) d/dx[ f(x)]
(the quotient rule)

d/dx [f(x) / g(x)] =
( g(x) d/dx[f(x)] - f(x) d/dx[g(x)] )

÷ [g(x)]^2
(the chain rule)

F'(x) =
f '(g(x)) • g'(x)

Leibniz notiation:
dy/dx = (dy/du) (du/dx)
(the chain rule & power rule)

d/dx [g(x)]^n =
n[g(x)]^(n-1) • g'(x)

Alternatively:
d/dx [u^n] = nu^(n-1) du/dx
Formal Definition of a Derivative

The tangent line to the curve y=f(x) at the pointP(a, f(a)) is the line through P with the slope...
m = lim(x→a) [(f(x) - f(a)] / [x - a]
Formal Definition of a Derivative Using H

The tangent line to the curve y=f(x) at the pointP(a, f(a)) with (h = x - a) & (x = a + h) is the line through P with the slope...
m = lim(h→0) [(f(a+h) - f(a)] / h