![d
= 1 ,
log bx x ln b
dx
x>0
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1
x>0
[ ln x ] = ,
dx
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1 du
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[ ln u] = × ,
dx
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Pg.244 deriving
Pg.244 deriving
formulas from
formulas from
definition
definition](https://image.slidesharecdn.com/4-131206203703-phpapp01/85/4-2-derivatives-of-logarithmic-functions-3-320.jpg)
4.2 derivatives of logarithmic functions
- 1. Derivatives of Logarithmic Functions John Napier (1550–1617), the inventor of logarithms
- 3. d = 1 , log bx x ln b dx x>0 d 1 x>0 [ ln x ] = , dx x d 1 du x>0 [ ln u] = × , dx u dx Pg.244 deriving Pg.244 deriving formulas from formulas from definition definition
- 4. Example: Find the derivative. ( ) f ( x ) = ln x 2 + 3 1 f ' ( x) = 2 ×2x ( x + 3) 2x f '( x) = 2 ( x + 3) Example: Find the derivative. x f ( x ) = ln x +1 1 1× x +1) −1×x ÷ ( f '( x) = × ÷ x ( x +1) 2 x +1 x +1 ( x +1) − x ÷ f '( x) = × 2 ÷ x ( x +1) f ' ( x) = x +1− x x ( x +1) 1 f '( x) = x ( x +1)
- 5. One more example !!! x 2 sin x f (x) = ln ÷ 1+ x Find the derivative of 1 2x sin x + (−cos x)×x 2 × 1+ x − ×x 2 sin x 1 2 1+ x × f '(x) = 2 2 x sin x 1+ x ÷ 1+ x ( ) x 2 sin x 2x sin x − x cos x) × 1+ x − 2 1+ x f '(x) = x 2 sin x 1+ x 2 ( )
- 6. Lets try this again Find the derivative of x 2 sin x f (x) = ln ÷ 1+ x 1 x 2 sin x 2 2 f (x) = ln ÷ = ln ( x sin x ) − ln 1+ x = ln x + lnsin x − ln ( 1+ x ) 2 1+ x therefore 1 f (x) = 2 ln x + ln ( sin x ) − ln ( 1+ x ) 2 so… 1 1 1 1 f '(x) = 2 × + ×cos x − × x sin x 2 1+ x 2 1 f '(x) = + cot x − x 2 ( 1+ x )
- 7. Logarithmic Differentiation The derivative of y= x 2 3 7x −14 ( 1+ x ) 2 4 is to messy to calculate directly so we first take the natural logarithm of both sides and use its properties. ( ln y = ln x 23 ) 7x −14 − ln ( 1+ x ) 2 4 1 3 = ln x + ln ( 7x −14) − ln ( 1+ x therefore 2 1 2 ln y = 2 ln x + ln ( 7x −14) − 4 ln ( 1+ x ) 3 ) 2 4
- 8. 1 dy 1 1 1 1 =2 + ×7 − 4 ×2x 2 y dx x 3 ( 7x −14) ( 1+ x ) 1 dy 2 7 8x = + − y dx x 3 ( 7x −14) ( 1+ x 2 ) 1 dy 2 7 8x ÷ y× = + − ×y 2 ÷ y dx x 3 ( 7x −14) ( 1+ x ) 23 dy 2 7 8x ÷ x 7x −14 = + − × 2 ÷ x 3 ( 7x −14) ( 1+ x ) 2 4 dx ( 1+ x )