notes-e-and-log-derivatives

# notes-e-and-log-derivatives - THEN y = ln x dy dx = 1 x y =...

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OPMT 5701 Notes on Natural Logarithm and the Exponential e 1. The Number e if y = e x then dy dx = e x if y = e f ( x ) then dy dx = e f ( x ) · f 0 ( x ) 2. Examples (a) y = e 3 x dy dx = e 3 x (3) (b) y = e 7 x 3 dy dx = e 7 x 3 (21 x 2 ) (c) y = e rt dy dt = re rt 3. Logarithm (Natural log) ln x (a) Rules of natural log If Then y = AB ln y =ln( AB )=ln A +ln B y = A/B ln y =ln A ln B y = A b ln y =ln( A b )= b ln A NOTE: ln( A + B ) 6 =ln A +ln B EVER!!! (b) derivatives IF
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Unformatted text preview: THEN y = ln x dy dx = 1 x y = ln ( f ( x )) dy dx = 1 f ( x ) Â· f ( x ) (c) Examples i. y = ln( x 2 âˆ’ 2 x ) dy/dx = 1 ( x 2 âˆ’ 2 x ) (2 x âˆ’ 2) ii. y = ln( x 1 / 2 ) = 1 2 ln x dy/dx = Î¼ 1 2 Â¶Î¼ 1 x Â¶ = 1 2 x 1...
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