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1. 1. Ejercicios de derivadas e integralesEste material puede descargarse desde http://www.uv.es/~montes/biologia/matcero.pdfDepartament d’Estad´ ıstica i Investigaci´ Operativa oUniversitat de Val`ncia e
2. 2. DerivadasReglas de derivaci´n o d Suma [f (x) + g(x)] = f (x) + g (x) dx d [kf (x)] = kf (x) dx Producto d [f (x)g(x)] = f (x)g(x) + f (x)g (x) dx d f (x) f (x)g(x) − f (x)g (x) Cociente = dx g(x) g(x)2 d {f [g(x)]} = f [g(x)]g (x) dx Regla de la cadena d {f (g[h(x)])} = f (g[h(x)])g [h(x)]h (x) dx d k d (x ) = kxk−1 [f (x)k ] = kf (x)k−1 f (x) dx dx d √ d 1/2 1 d f (x) Potencia ( x) = (x ) = √ [ f (x)] = dx dx 2 x dx 2 f (x) d 1 d −1 1 d 1 f (x) = (x ) = − 2 =− dx x dx x dx f (x) f (x)2
3. 3. 2Reglas de derivaci´n (continuaci´n) o o d d (sin x) = cos x [sin f (x)] = cos f (x)f (x) dx dx d d Trigonom´tricas e (cos x) = − sin x [cos f (x)] = − sin f (x)f (x) dx dx d d (tan x) = 1 + tan2 x [tan f (x)] = [1 + tan2 f (x)]f (x) dx dx d 1 d f (x) (arcsin x) = √ [arcsin f (x)] = dx 1 − x2 dx 1 − f (x)2 d −1 d −f (x) Funciones de arco (arc cos x) = √ [arc cos f (x)] = dx 1 − x2 dx 1 − f (x)2 d 1 d f (x) (arctan x) = [arctan f (x)] = dx 1 + x2 dx 1 + f (x)2 d x d f (x) (e ) = ex (e ) = ef (x) f (x) dx dx Exponenciales d x d f (x) (a ) = ax ln a (a ) = af (x) ln af (x) dx dx d 1 d f (x) (ln x) = (ln f (x)) = dx x dx f (x) Logar´ ıtmicas d 1 1 d f (x) 1 (lg x) = (lg f (x)) = dx a x ln a dx a f (x) ln a