Basic Differentiation and Rates of Change

# Basic Differentiation and Rates of Change - x x dx d sin...

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Basic Differentiation and Rates of Change Section 2.2

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Theorem 2.2 The Constant Rule The derivative of a constant function is 0. That is, if c is a real number, then [ ] 0 = c dx d
Theorem 2.3 The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and For f to be differentiable at x = 0, n must be a real # such that x n-1 is defined on an interval containing 0. [ ] 1 - = n n nx x dx d

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In the Power Rule, the case for which n = 1 is best thought of as a separate differentiation rule. That is, [ ] 1 = x dx d
Theorem 2.4 The Constant Multiple Rule If f is a differentiable function and c is a real #, then cf is also differentiable and [ ] ) ( ' ) ( x cf x cf dx d =

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Theorem 2.5 The Sum and Difference Rules The derivative of the sum (or difference) of two differentiable functions is differentiable and is the sum (or difference) of their derivatives. [ ] ) ( ' ) ( ' ) ( ) ( x g x f x g x f dx d ± = ±
Theorem 2.6 Derivatives of Sine and Cosine Functions [ ] x x dx d cos sin = [ ]

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Unformatted text preview: x x dx d sin cos-= Average Velocity t s time in change distance in change ∆ ∆ = Velocity If s = s(t) is the position function for an object moving along a straight line. The velocity of the object at time t is Velocity is the derivative of the position function. Velocity can be negative, positive, or zero. ) ( ' ) ( ) ( lim ) ( t s t t s t t s t v x = ∆-∆ + = → ∆ Speed The speed of an object is the absolute value of its velocity. Speed cannot be negative. where s is the initial height of the object, v is the initial velocity and g is the acceleration due to gravity. 2 2 1 ) ( s t v gt t s + + = Joke Time Why was Cinderella such a bad baseball player? She ran away from the ball! What did Snow White say when the snapshots she’d ordered didn’t arrive? Someday my prints will come!...
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Basic Differentiation and Rates of Change - x x dx d sin...

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