Finding the derivatives of functions

[lamerali]

Hi, I'm working with finding the derivatives of functions, which I'm not very comfortable with; if someone could please check my answers to the following questions i would be VERY grateful! Thank you! :)

find the derivative of the following function:

Question 1:

y = \frac{ 2^{x} }{ e^{x} }

My Answer

y1 = \frac{ e^{x} . ln2 . 2^{x} + 2^{x} . e^{x} }{ e^{x}^{2} }
= \frac{ 2^{x} (ln2 + 1) }{ e^{x} }

Question 2:

f(x) = 2x ln(x^{2} + 5)

My answer

f ^{1} (x) = 2ln(x^{2} + 5) + (2x) . \frac{1}{x^{2} + 5} . (2x)

= 2 ln(x^{2} + 5) + \frac{4x^{2}}{x^{2} + 5}

Question 3:

g(x) = \frac{ln x}{e^{x}^{2} + 2}

My answer:

g^{1}(x) = \frac{(e^{x}^{2} + 2) . (1/x) - lnx . 2xe^{x}^{2}}{(e^{x}^{2} + 2)^{2}}

= \frac{\frac{e^{x}^{2} + 2}{x} - lnx . 2xe^{x}^{2}}{(e^{x}^{2} + 2)^{2}}

for the last two questions I'm not sure if i simplified enough...if anyone could guide me in the right direction where needed i'd really appreciate it! thanks in advance!

 

[HallsofIvy]

Hi, I'm working with finding the derivatives of functions, which I'm not very comfortable with; if someone could please check my answers to the following questions i would be VERY grateful! Thank you! :)

find the derivative of the following function:

Question 1:

y = \frac{ 2^{x} }{ e^{x} }

My Answer

y1 = \frac{ e^{x} . ln2 . 2^{x} + 2^{x} . e^{x} }{ e^{x}^{2} }

Almost right you need "-" , not "+" in the numerator- quotient rule: (u/v)'= (u'v- uv')/v^2.

= \frac{ 2^{x} (ln2 + 1) }{ e^{x} }

Question 2:

f(x) = 2x ln(x^{2} + 5)

My answer

f ^{1} (x) = 2ln(x^{2} + 5) + (2x) . \frac{1}{x^{2} + 5} . (2x)

= 2 ln(x^{2} + 5) + \frac{4x^{2}}{x^{2} + 5}

Yes, that looks good.

Question 3:

g(x) = \frac{ln x}{e^{x}^{2} + 2}

My answer:

g^{1}(x) = \frac{(e^{x}^{2} + 2) . (1/x) - lnx . 2xe^{x}^{2}}{(e^{x}^{2} + 2)^{2}}

= \frac{\frac{e^{x}^{2} + 2}{x} - lnx . 2xe^{x}^{2}}{(e^{x}^{2} + 2)^{2}}

for the last two questions I'm not sure if i simplified enough...if anyone could guide me in the right direction where needed i'd really appreciate it! thanks in advance!

The third problem also looks good to me.

 

[lamerali]

Great! Thank you HallsofIvy! :D