find f''(x). f(x)= x/ 5+e^x

Question

can you work it out please

Answer 1

1) Rewrite f(x) so it would be visually easier to see.

f(x) = x * (5+e^x)

2) As you can see this form looks like f(x)= g(x)*h(x)

h(x) = 5 + e^x

g(x) = x

apply rule if f(x)= g(x)*h(x) then f'(x) = g'(x)*h(x) + h'(x)*g(x)

h'(x) = e^x

g'(x) = 1

3) sub new values for f'(x) = (1)*(e^x) + (e^x)*(x) = e^x [ 1+x ]

f'(x) = e^x [ 1+x ]

4) As you can see this form looks like f(x)= g(x)*h(x)

g(x) = e^x ........ g'(x) = e^x

h(x) = 1+x ....... h'(x) = 1

5) sub new values

f''(x) = e^x [ 1+x ] + (1*e^x)

= e^x + x*e^x + e^x

= xe^x + 2e^x

= e^x ( x + 2 )

---> f''(x) = e^x ( x + 2 )

If you need any help let schedule a lesson :)

Answer 2

This is a quotient rule problem.

f(x)=x/(5+e^x)

f ' (x) = (5+e^x)(1) - x(e^x) = (5+e^x)- x(e^x)

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(5+e^x)² (5+e^x)²

f " (x) =[ (5+e^x)²((e^x) -((1)(e^x)+x(e^x))] - [ ((5+e^x) - x(e^x)) (2)(5+e^x)(e^x)]

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(5+e^x)⁴

The above answer could be simplified further but it is not necessary to do so.