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Notes_Exercises_Set05

# Notes_Exercises_Set05 - 1 The exponential and logarithmic...

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1. The exponential and logarithmic functions (Inverse Functions): ) ln( u p e u p = = Example: 0 ) 1 ln( 1 0 = = e ) ln( ) ln( u p e u and p e = = Example: 1 ) ln( = e Rules of Exponents Rules of Logarithms 2 1 2 1 2 1 2 1 2 1 2 1 ) ( / p p p p p p p p p p p p e e e e e e e e = = = - + ) ln( 1 ln[ ) ln( ) ln( ] / ln[ ) ln( ) ln( ] ln[ u v u v u v u v u uv v = - = + = 1 0 = e 0 ) 1 ln( = ) ln( a u u e a = Conversion to base e ) ln( ) ln( log a u u a = Conversion to base e 2. Derivatives x x x e e D = and ) ln( ) ln( ) ln( ) ln( a a a e e D a D x a x a x x x x = = = Anti-derivative: c e dx e x x + = ° and c a a dx a x x + = ° ) ln( ) exp( * * * ] 1 )[ exp( )] 0 exp( ) 0 [exp( lim ) exp( ] 1 ) [exp( lim ) exp( ] 1 ) )[exp( exp( lim )] exp( ) exp( ) exp( lim ) exp( ) exp( lim ] [ 0 0 0 0 0 x x x x x x x x x x x x x x x x x x x e D x x x x x x x = = Δ - Δ + = Δ - Δ = Δ - Δ = = Δ - Δ = Δ - Δ + = Δ Δ Δ Δ Δ *** since 1 )] 0 exp( ) 0 [exp( lim 0 = Δ - Δ + Δ x x x because we define exp(x) with a special base = e , such that 1 is the value of the slope of the tangent line at (0,1) for the exponential function. 3. Derivative of ln(x) x x D x 1 ) ln( = => c x dx x + = ° | | ln 1 ± ± ² ³ ´ ´ µ = = ) ln( 1 1 ) ln( ) ln( ) ( log a x a x D x D x a x Let u = ln(x) => x = e u Differentiating wrt x: x dx du dx du e u 1 1 = · = Therefore: x x D x 1 ) ln( =

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4. Additional Formulas: ) tan( ) sin ( cos 1 ) ln(cos x x x x D x - = - = c x c x dx x + = + - = ° | sec | ln | cos | ln ) tan( ) cot( ) (cos sin 1 ) ln(sin x x x x D x = = c x dx x + = ° | sin |
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Notes_Exercises_Set05 - 1 The exponential and logarithmic...

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