Calculus

Section 2.3The Product and Quotient

Rules and Higher-Order Derivatives

The Product Rule

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f (x)g(x) f (x) g (x) g(x) f (x)

The product of two differentiable functions is differentiable.

The first times the derivative of the second plus the second times the derivative of the first.

The Quotient Rule

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f (x)g(x)

g(x) f (x) f (x) g (x)g(x) 2

The quotient of two differentiable functions is differentiable at all values of x such that g(x) doesn’t equal zero.

Speak Like a Pirate: Calculus

Lodhi minus hidlo, Square the denominator,

And you’re good to go!-Kwo Shunt Ruel-

Speak Like A Pirate Day

Proof of The Derivative of Tangent

Consider tan x sin xcos x

Apply the Quotient Rule

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tan x cos x(cos x) sin x( sin x)cos x 2

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tan x cos2 x sin2 xcos2 x

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tan x 1cos2 x

sec2 x

Derivatives of Trig Functions

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sin x cos x ddx

csc x csc x cot x

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cos x sin x ddx

sec x sec x tan x

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tan x sec2 x ddx

cot x csc2 x

Proof of The Other ThreeNow, prove the other three trigonometric functions:

Textbook p. 125 # 81

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csc x csc x cot x ddx

sec x sec x tan x

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cot x csc2 x