Question

Se F(x)=ln (x3+1), é correto afirmar que:


a.
F′′(x)= 6x4+9x2+6x
x5+2x3+1

b.
F′′(x)= x5+2x3+1
x4+9x2+6x

c.
F′′(x)= x5+2x3+1
x4+2x2+6x

d.
F′′(x)= x4+9x2+6x
x5+2x3+1

e.
F′′(x)= −3x4+6x
x6+2x3+1

Answer 1

Sejam u e v duas funções:

• $\frac{d}{dx}\:\ln u=\frac{u'}{u}$

• $\frac{d}{dx}\:\frac{u}{v}=\frac{u'v-uv'}{v^2}$

$F(x)=\ln(x^3+1)\\\\F'(x)=\frac{(x^3+1)'}{x^3+1}\\\\F'(x)=\frac{3x^2}{x^3+1}\\\\\\F''(x)=\frac{(3x^2)'(x^3+1)-(3x^2)(x^3+1)'}{(x^3+1)^2}\\\\F''(x)=\frac{6x(x^3+1)-(3x^2)3x^2}{x^6+2x^3+1}\\\\F''(x)=\frac{-3x^4+6x}{x^6+2x^3+1}$