Notes on the Calculus of the Natural Exponential Function

Notes on the Calculus of the Natural Exponential Function -...

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Unformatted text preview: Notes on the Calculus of the Natural Exponential Function This lesson concerns the differentiation and integration of functions involving the natural exponential function. Differentiation : Consider the function ( 29 x f x e = . We wish to calculate ( 29 f x ′ . To do so, let x y e = and use logarithmic differentiation: 1 ln ln 1 x d d dy y e y x y x dx dx y dx = ⇔ = ⇔ = ⇔ = x x dy d y e e dx dx ⇔ = ⇔ = . Summarizing, x x d e e dx = for all values of the variable x . ( 29 ( 29 ( 29 g x g x d e e g x dx ′ = whenever ( 29 g x ′ exists by the Chain Rule. Integration : All differentiation formulas give rise to integration formulas. Since x x d e e dx = , the following integration formulas also hold true: x x e dx e C = + ∫ ( 29 ( 29 ( 29 g x g x e g x dx e C ′ = + ∫ Comments on these formulas : There are several ideas to keep in mind when differentiating or integrating functions involving the natural exponential function. • The derivative of ( 29 g x e is just the function ( 29 g x e times the derivative of the exponent. Don’t forget to multiply by the derivative of the exponent! • Whenever one is integrating ( 29 g x e always use substitution with the choice of u being the exponent. Example 1 : Calculate sin cos x x d e dx- Solution : ( 29 sin cos sin cos sin cos x x x x d d e e x x dx dx-- =- ( From above with ( 29 sin cos g x x x =- ) ( 29 sin cos cos sin x x e x x- =- - ( 29 sin cos cos sin x x e x x- = + . Example 2 : Find dy dt where ( 29 2 ln 3 t y e t = Solution : ( Always remember the previous rules for differentiation. ) ( 29 ( 29 ( 29 2 2 2 ln 3 ln 3 ln 3 t t t dy d d d e t e t t e dt dt dt dt = = + ( 29 ( 29 ( 29 2 2 1 3 ln 3 2 3 t t d d e t t e t t dt dt = + ( 29 ( 29 ( 29 2 2 1 3 ln 3 2...
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Notes on the Calculus of the Natural Exponential Function -...

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