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Ca´lculo Diferencial e Integral I Regras Operato´rias das Derivadas Luiz C. M. de Aquino aquino.luizclaudio@gmail.com http://sites.google.com/site/lcmaquino http://www.youtube.com/LCMAquino Regras Operato´rias das Derivadas Introduc¸a˜o Tabela ba´sica de derivadas f (x) f ′(x) c 0 xn nxn−1 n √ x 1 n n √ xn−1 ax ax ln a loga x 1 x ln a sen x cos x cos x − sen x Regras Operato´rias das Derivadas Teorema Sejam f e g func¸o˜es diferencia´veis em um mesmo dom´ınio D. Sa˜o va´lidas as afirmac¸o˜es: (i) [cf (x)]′ = cf ′(x), com c uma constante real qualquer; (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x); (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x); (iv) [ f (x) g(x) ]′ = f ′(x)g(x)−f (x)g ′(x) [g(x)]2 , com g(x) 6= 0 no dom´ınio D. Observac¸a˜o De (i) e (ii) segue que: (v) [f (x)− g(x)]′ = f ′(x)− g ′(x). Regras Operato´rias das Derivadas Teorema Sejam f e g func¸o˜es diferencia´veis em um mesmo dom´ınio D. Sa˜o va´lidas as afirmac¸o˜es: (i) [cf (x)]′ = cf ′(x), com c uma constante real qualquer; (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x); (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x); (iv) [ f (x) g(x) ]′ = f ′(x)g(x)−f (x)g ′(x) [g(x)]2 , com g(x) 6= 0 no dom´ınio D. Observac¸a˜o De (i) e (ii) segue que: (v) [f (x)− g(x)]′ = f ′(x)− g ′(x). Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Aplicando a definic¸a˜o de derivada, temos que [cf (x)]′ = lim h→0 cf (x + h)− cf (x) h = lim h→0 c[f (x + h)− f (x)] h = c lim h→0 f (x + h)− f (x) h = cf ′(x) Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Aplicando a definic¸a˜o de derivada, temos que [cf (x)]′ = lim h→0 cf (x + h)− cf (x) h = lim h→0 c[f (x + h)− f (x)] h = c lim h→0 f (x + h)− f (x) h = cf ′(x) Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Aplicando a definic¸a˜o de derivada, temos que [cf (x)]′ = lim h→0 cf (x + h)− cf (x) h = lim h→0 c[f (x + h)− f (x)] h = c lim h→0 f (x + h)− f (x) h = cf ′(x) Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Exemplo 1: Seja a func¸a˜o f (x) = 5x3. f ′(x) = ( 5x3 )′ = 5 ( x3 )′ = 5 ( 3x2 ) = 15x2 Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Exemplo 1: Seja a func¸a˜o f (x) = 5x3. f ′(x) = ( 5x3 )′ = 5 ( x3 )′ = 5 ( 3x2 ) = 15x2 Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Exemplo 1: Seja a func¸a˜o f (x) = 5x3. f ′(x) = ( 5x3 )′ = 5 ( x3 )′ = 5 ( 3x2 ) = 15x2 Regras Operato´rias das Derivadas (i) [cf (x)]′ = cf ′(x) Exemplo 1: Seja a func¸a˜o f (x) = 5x3. f ′(x) = ( 5x3 )′ = 5 ( x3 )′ = 5 ( 3x2 ) = 15x2 Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x) + g(x)]′ = lim h→0 [f (x + h) + g(x + h)]− [f (x) + g(x)] h = lim h→0 [f (x + h)− f (x)] + [g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h + lim h→0 g(x + h)− g(x) h = f ′(x) + g ′(x) Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x) + g(x)]′ = lim h→0 [f (x + h) + g(x + h)]− [f (x) + g(x)] h = lim h→0 [f (x + h)− f (x)] + [g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h + lim h→0 g(x + h)− g(x) h = f ′(x) + g ′(x) Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x) + g(x)]′ = lim h→0 [f (x + h) + g(x + h)]− [f (x) + g(x)] h = lim h→0 [f (x + h)− f (x)] + [g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h + lim h→0 g(x + h)− g(x) h = f ′(x) + g ′(x) Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x . f ′(x) = (2x + cos x)′ = (2x)′ + (cos x)′ = 2x ln 2− sen x Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x . f ′(x) = (2x + cos x)′ = (2x)′ + (cos x)′ = 2x ln 2− sen x Regras Operato´rias das Derivadas (ii) [f (x) + g(x)]′ = f ′(x) + g ′(x) Exemplo 2: Seja a func¸a˜o f (x) = 2x + cos x . f ′(x) = (2x + cos x)′ = (2x)′ + (cos x)′ = 2x ln 2− sen x Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Aplicando a definic¸a˜o de derivada, temos que [f (x)g(x)]′ = lim h→0 f (x + h)g(x + h)− f (x)g(x) h = lim h→0 f (x + h)g(x + h)− f (x)g(x) + f (x)g(x + h)− f (x)g(x + h) h = lim h→0 [f (x + h)− f (x)]g(x + h) + f (x)[g(x + h)− g(x)] h = lim h→0 [f (x + h)− f (x)]g(x + h) h + lim h→0 f (x)[g(x + h)− g(x)] h = lim h→0 f (x + h)− f (x) h lim h→0 g(x + h) + f (x) lim h→0 g(x + h)− g(x) h = f ′(x)g(x) + f (x)g ′(x) Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x . f ′(x) = (ln x sen x)′ = (ln x)′ sen x + ln x ( sen x)′ = 1 x sen x + ln x cos x Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x . f ′(x) = (ln x sen x)′ = (ln x)′ sen x + ln x ( sen x)′ = 1 x sen x + ln x cos x Regras Operato´rias das Derivadas (iii) [f (x)g(x)]′ = f ′(x)g(x) + f (x)g ′(x) Exemplo 3: Seja a func¸a˜o f (x) = ln x sen x . f ′(x) = (ln x sen x)′ = (ln x)′ sen x + ln x ( sen x)′ = 1 x sen x + ln x cos x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Aplicando a definic¸a˜o de derivada, temos que [ f (x) g(x) ]′ = lim h→0 f (x+h) g(x+h) − f (x)g(x) h = lim h→0 f(x + h)g(x)− f (x)g(x + h) hg(x + h)g(x) = lim h→0 f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x) hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)] hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Aplicando a definic¸a˜o de derivada, temos que [ f (x) g(x) ]′ = lim h→0 f (x+h) g(x+h) − f (x)g(x) h = lim h→0 f (x + h)g(x)− f (x)g(x + h) hg(x + h)g(x) = lim h→0 f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x) hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)] hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Aplicando a definic¸a˜o de derivada, temos que [ f (x) g(x) ]′ = lim h→0 f (x+h) g(x+h) − f (x)g(x) h = lim h→0 f (x + h)g(x)− f (x)g(x + h) hg(x + h)g(x) = lim h→0 f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x) hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)] hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Aplicando a definic¸a˜o de derivada, temos que [ f (x) g(x) ]′ = lim h→0 f (x+h) g(x+h) − f (x)g(x) h = lim h→0 f (x + h)g(x)− f (x)g(x + h) hg(x + h)g(x) = lim h→0 f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x) hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)] hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Aplicando a definic¸a˜o de derivada, temos que [ f (x) g(x) ]′ = lim h→0 f (x+h) g(x+h) − f (x)g(x) h = lim h→0 f (x + h)g(x)− f (x)g(x + h) hg(x + h)g(x) = lim h→0 f (x + h)g(x)− f (x)g(x + h) + f (x)g(x)− f (x)g(x) hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x)− f (x)[g(x + h)− g(x)] hg(x + h)g(x) = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ [ f (x) g(x) ]′ = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) = f ′(x)g(x) [g(x)]2 − f (x)g ′(x) [g(x)]2 = f ′(x)g(x)− f (x)g ′(x) [g(x)]2 Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ [ f (x) g(x) ]′ = lim h→0 [f (x + h)− f (x)]g(x) hg(x + h)g(x) − lim h→0 f (x)[g(x + h)− g(x)] hg(x + h)g(x) = f ′(x)g(x) [g(x)]2 − f (x)g ′(x) [g(x)]2 = f ′(x)g(x)− f (x)g ′(x) [g(x)]2 Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (iv) [ f (x) g(x) ]′ Exemplo 4: Seja a func¸a˜o f (x) = tg x . f ′(x) = ( tg x)′ = ( sen x cos x )′ = ( sen x)′ cos x − sen x(cos x)′ [cos x ]2 = cos x cos x − sen x(− sen x) cos2 x = cos2 x + sen 2x cos2 x = 1 cos2 x = sec2 x Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) [f (x)− g(x)]′ = [f (x) + (−1)g(x)]′ = f ′(x) + [(−1)g(x)]′ = f ′(x) + (−1)[g(x)]′ = f ′(x)− g ′(x) Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) [f (x)− g(x)]′ = [f (x) + (−1)g(x)]′ = f ′(x) + [(−1)g(x)]′ = f ′(x) + (−1)[g(x)]′ = f ′(x)− g ′(x) Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) [f (x)− g(x)]′ = [f (x) + (−1)g(x)]′ = f ′(x) + [(−1)g(x)]′ = f ′(x) + (−1)[g(x)]′ = f ′(x)− g ′(x) Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) [f (x)− g(x)]′ = [f (x) + (−1)g(x)]′ = f ′(x) + [(−1)g(x)]′ = f ′(x) + (−1)[g(x)]′ = f ′(x)− g ′(x) Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x . f ′(x) = ( x2 −√x)′ = ( x2 )′ − (√x)′ = 2x − 1 2 √ x Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x . f ′(x) = ( x2 −√x)′ = ( x2 )′ − (√x)′ = 2x − 1 2 √ x Regras Operato´rias das Derivadas (v) [f (x)− g(x)]′ = f ′(x)− g ′(x) Exemplo 5: Seja a func¸a˜o f (x) = x2 −√x . f ′(x) = ( x2 −√x)′ = ( x2 )′ − (√x)′ = 2x − 1 2 √ x
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