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Hall's Review 3.2

Final Review

Define a tangent line The tangent line to the curve y = f(x) at the point P(c, f(c)) is that line through P with slope m= lim (as h approaches 0) (f(c+h)- f(c))/h proivided the limit exists and is not infiniety or negative infinity.
Define derivative The derivative of a function f is another function f’ whose value at any number x is f'(x) = lim(as h approaches 0) [f(x+h)- f(x)]/h
Thm. differenciability implies what? continuity; If f’(c) exist, the f is continuous at c
Thm: Constant Function Rule If f(x) = k, where k is a constant, then for any x, f’(x) = 0
Thm: Identity Function Rule If f(x) = x then f’(x) = 1
Thm: Power Rule If f(x) = x^n, where n is a positive integer, the f’(x) = nx^n-1; that is Dx(x^n) = nx^n-1
Thm: Constant Multiple Rule If k is a constant and f is a differentiable function, then (kf)’(x) = k * f’(x)
Thm: Sum Rule If f and g are differentiable functions, then (f+g)’(x) = f’(x) + g’(x)
Thm: Difference Rule If f and g are differentiable functions, then (f-g)’(x) = f’(x) - g’(x)
Thm: Product Rule If f and g are differentiable functions, then (f*g)’(x) = f(x)g’(x) + g(x)f’(x)
Thm: Quotient Rule Let f and g be differentiable functions with g(x) does not equal zero. Then (f/g)'(x) = [g(x)f'(x)-g'(x)f(x)]/(g(x))^2
Derivative of sin x cos x
Derivative of cos x -sin x
Derivative of tan x (sec x)^2
Derivative of cot x -(csc x)^2
Derivative of sec x sec x tan x
Derivative of csc x -csc x cot x
Thm. Chain Rule Let y = f(u) and u = g(x). If g is differentiable at x and f is differentiable at u = g(x), then the composite function f(g(x)) is differentiable at x and f(g(x))' = f'(g(x))g'(x)
Created by: agea