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LogDerivatives

# LogDerivatives - iversitas Un DEO PATRI Derivatives of...

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U n i v e r s i t a s S a s k a t c h e w a n e n s i s DEO ET PAT- RIÆ 2001 Doug MacLean Derivatives of Exponentials & Logarithms d dx ( e x ) = e x and d dx ( a x ) = ( ln a)a x d dx ( ln x ) = 1 x and d dx ( log a x ) = 1 ( ln a)x We may get more general formulas by using the Chain Rule: d dx e f (x) = e f (x) f (x) and d dx a f (x) = ( ln a)a f (x) f (x) d dx ( ln f (x) ) = f (x) f (x) and d dx ( log a f (x) ) = f (x) ( ln a)f (x) = 1 ln a f (x) f (x) The quantity f (x) f (x) is called the relative rate of change of the function f , and is very important in practical applications. Example 1: Find the derivative of y = e 3 x . Solution: y = e 3 x ( 3 x) = e 3 x 3 = 3 e 3 x Example 2: Find the derivative of y = e x 3 . Solution: y = e x 3 (x 3 ) = e x 3 ( 3 x 2 ) = 3 x 2 e x 3 1

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Example 3: Find the derivative of y = 3 x 2 . Solution: y = ( ln3 ) 3 x 2 (x 2 ) = ( ln3 ) 3 x 2 ( 2 x) = ( 2 x ln3 ) 3 x 2 Example 4: Find the derivative of y = ln (x 5 + x 3 + 1 ) . Solution: y = 1 x 5 + x 3 + 1 (x 5 + x 3 + 1 ) = 1 x 5 + x 3 + 1 ( 5 x 4 + 3 x 2 ) = x 2 ( 5 x 2 + 3 ) x 5 + x 3 + 1 Logarithmic Differentiation
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LogDerivatives - iversitas Un DEO PATRI Derivatives of...

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