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 Universidad Universidad Autónoma de Barcelona (UAB) Grado Ciencias Ambientales - 1º curso Asignatura Mates Año del apunte 2009 Páginas 2 Fecha de subida 25/05/2014 Descargas 0 Puntuación media

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Matem` atiques 23816 Ci` encies Ambientals Teoria LLISTA DE DERIVADES DE LES FUNCIONS ELEMENTALS 1. (xn ) = n xn−1 , 2. (ex ) = ex , 3. (ax ) = ax ln a, 4. (loga x) = 5. (ln x) = x1 , 6. (sin x) = cos x, 7. (cos x) = − sin x, 8. (tan x) = sec2 x = 1 + tan2 x = 9. (arcsin x) = √ 1 , 1−x2 11. (arctan x) = 1 x ln a , 10. (arccos x) = 1 , cos2 x √ −1 , 1−x2 1 .
1+x2 Utilitzant la regla de la cadena, (g ◦ f ) = (g ◦ f ) · f , i les derivades de les funcions elementals que acabem de veure tenim el seg¨ uent: 1. ((f (x))n ) = n (f (x))n−1 · f (x), 2. (ef (x) ) = ef (x) · f (x), 3. (af (x) ) = af (x) ln a f (x), 4. (loga f (x)) = 5. (ln f (x)) = f (x) f (x) , f (x) f (x) ln a , 6. (sin f (x)) = f (x) · cos(f (x)), 7. (cos f (x)) = −f (x) · sin(f (x)), 8. (tan f (x)) = sec2 (f (x)) f (x) = = (1 + tan2 f (x))f (x) = 9. (arcsin f (x)) = √ 11. (arctan f (x)) = f (x) , 1−(f (x))2 f (x) .
1+(f (x))2 10. (arccos f (x)) = √ −f f (x) , cos2 f (x) (x) , 1−(f (x))2 EXEMPLES (a) f (x) = f (x) = (b) f (x) = x2 −5x ; x3 −1 (2x − 5)(x3 − 1) − (x2 − 5x)3x2 2x4 − 2x − 5x3 + 5 − 3x4 + 15x3 −x4 + 10x3 − 2x + 5 = = .
(x3 − 1)2 (x3 − 1)2 (x3 − 1)2 √ x2 + 3; 1 f (x) = ((x2 + 3) 2 ) = −1 1 1 2 x (x + 3) 2 −1 (2x) = (x2 + 3) 2 x = √ .
2 2 x +3 (c) f (x) = (x2 − 7)(x3 − 12x + 2); f (x) = 2x(x3 −12x+2)+(x2 −7)(3x2 −12) = 2x4 −24x2 +4x+3x4 −12x2 −21x2 +84 = 5x4 −57x2 +4x+84.
(d) f (x) = ln x x ; f (x) = (e) f (x) = log3 (x + √ 1 x x − ln x 1 − ln x = .
2 x x2 x2 − 1); 1 1 1 + √xx2 −1 1 + 21 (x2 − 1) 2 −1 2x 1 1 + (x2 − 1)− 2 x 1 1 √ √ √ f (x) = = = = · · · 2 2 2 ln 3 ln 3 x+ x −1 x+ x −1 x + x − 1 ln 3 = x √ x2 −1+x √ x2 −1 √ + x2 − √ 1 x2 − 1 + x 1 x √ √ √ · · .
= = 2 2 1 ln 3 (x + x − 1) x − 1 ln 3 ln 3 x2 − 1 (f) f (x) = x3 · e−3x ; f (x) = 3x2 · e−3x + x3 · e−3x (−3) = 3x2 · e−3x − 3x3 · e−3x = 3x2 e−3x (1 − x).
(g) f (x) = cos(3x2 + 4x − 1); f (x) = − sin(3x2 + 4x − 1) · (6x + 4).
(h) f (x) = arcsin 1 ln x f (x) = ; − x1 · (ln x)2 1 1− 1 2 ln x − x1 = (ln x)2 (ln x)2 −1 (ln x)2 =− 1 x ln x (ln x)2 − 1 .
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