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1. 1. Ejercicios de derivadas e integrales Este material puede descargarse desde http://www.uv.es/~montes/biologia/matcero.pdf Departament d’Estad´ıstica i Investigaci´o Operativa Universitat de Val`encia
2. 2. Derivadas Reglas de derivaci´on Suma d dx [f(x) + g(x)] = f (x) + g (x) d dx [kf(x)] = kf (x) Producto d dx [f(x)g(x)] = f (x)g(x) + f(x)g (x) Cociente d dx f(x) g(x) = f (x)g(x) − f(x)g (x) g(x)2 d dx {f[g(x)]} = f [g(x)]g (x) Regla de la cadena d dx {f(g[h(x)])} = f (g[h(x)])g [h(x)]h (x) d dx (xk ) = kxk−1 d dx [f(x)k ] = kf(x)k−1 f (x) Potencia d dx ( √ x) = d dx (x1/2 ) = 1 2 √ x d dx [ f(x)] = f (x) 2 f(x) d dx 1 x = d dx (x−1 ) = − 1 x2 d dx 1 f(x) = − f (x) f(x)2
3. 3. 2 Reglas de derivaci´on (continuaci´on) d dx (sin x) = cos x d dx [sin f(x)] = cos f(x)f (x) Trigonom´etricas d dx (cos x) = − sin x d dx [cos f(x)] = − sin f(x)f (x) d dx (tan x) = 1 + tan2 x d dx [tan f(x)] = [1 + tan2 f(x)]f (x) d dx (arcsin x) = 1 √ 1 − x2 d dx [arcsin f(x)] = f (x) 1 − f(x)2 Funciones de arco d dx (arc cos x) = −1 √ 1 − x2 d dx [arc cos f(x)] = −f (x) 1 − f(x)2 d dx (arctan x) = 1 1 + x2 d dx [arctan f(x)] = f (x) 1 + f(x)2 d dx (ex ) = ex d dx (ef(x) ) = ef(x) f (x) Exponenciales d dx (ax ) = ax ln a d dx (af(x) ) = af(x) ln af (x) d dx (ln x) = 1 x d dx (ln f(x)) = f (x) f(x) Logar´ıtmicas d dx (lga x) = 1 x 1 ln a d dx (lga f(x)) = f (x) f(x) 1 ln a