Vol2 Derivadas, aplicaciones y temas especiales - CIMM

38 Elementos de cálculo, volumen 2
(c)Tenemos:
(d)
y ′ =
=
=
=
y ′ =
=
=
=
′
x + ln x
=
x + 2
(x + ln x) ′ (x + 2) − (x + ln x)(x + 2) ′
(x + 2) 2
(1 + 1
x )(x + 2) − (x + ln x)
(x + 2) 2
3x + 2 − x ln x
x(x + 2) 2
′
x + 3 ln(x − 2)
[x + 3 ln(x − 2)] ′
2 x + 3 ln(x − 2)
1 + 3 1
x−2
2 x + 3 ln(x − 2)
x + 1
2(x − 2) x + 3 ln(x − 2)
• Determine la derivada de las siguientes funciones:
(a) y = xe x + x 2 (b) y = x2 + 1
(d) y = √ x + 3e x
e x − 3
Solución: (a) y ′ = (xe x + x 2 ) ′ = (xe x ) ′ + (x 2 ) ′
(b) y ′ =
= (x) ′ e x + x(e x ) ′ + 2x
= 1 · e x + x · e x + 2x = e x (1 + x) + 2x
x 2 + 1
e x − 3
′
(c) y = e x2 +2x−3
= (x2 + 1) ′ (e x − 3) − (x 2 + 1)(e x − 3) ′
(e x − 3) 2
= 2x(ex − 3) − (x 2 + 1)e x
(e x − 3) 2
(c) y ′ = e x2 +2x−3 ′ x
= e 2 +2x−3 2 ′ x
(x + 2x − 3) = e 2 +2x−3
(2x + 2)
(d) y ′ √x ′
= + 3ex = (x + 3ex ) ′
2 √ 1 + 3ex
=
x + 3ex 2 √ x + 3ex