Function of a Function Rule+Exponential (1 Viewer)

[Avenger6]

Hi, I have two questions here that really puzzled me, they both involve using the Function of a Function Rule with regard to exponential functions, both times I seem to get the incorrect answer so a step-by-step explaination would be greatly appreciated.

1) Differentiate e^2x+1/2x+5

2) Find the second derivative of (e^2x+1)⁷

Help is much appreciated .

 

[veloc1ty]

I gave it a shot, someone better double-check my answers though.

 

[Avenger6]

Thanks for the quick response. The first question is correct however the answer the back of the book for the second question is:

168e^4x(e^2x+1)⁵+28e^2x(e^2x+1)⁶
= 28e^2x(e^2x+1)⁵(7e^2x+1)

I originally had the same answer you gave, could the answer in the back be incorrect, I can not see how they have solved it???

 

[Slidey]

Hi, I have two questions here that really puzzled me, they both involve using the Function of a Function Rule with regard to exponential functions, both times I seem to get the incorrect answer so a step-by-step explaination would be greatly appreciated.

1) Differentiate e^2x+1/2x+5

2) Find the second derivative of (e^2x+1)⁷

Help is much appreciated .

y=e^(2x+1)/(2x+5)
lny=2x+1-ln(2x+5)
y'/y=2-2/(2x+5)
y'=2y(2x+4)/(2x+5)
y'=4(x+2)e^(2x+1)/(2x+5)^2

Yep, that's correct, velocity.

2) Find the second derivative of (e^2x+1)⁷

y=(e^2x + 1)^7
y'=14e^2x(e^2x+1)^6
y''=168e^4x(e^2x+1)^5 + 28e^2x(e^2x+1)^6 (product rule)
y''=28e^2x(e^2x+1)^5(7e^2x+1)

 

[Avenger6]

y=(e^2x + 1)^7
y'=14e^2x(e^2x+1)^6
y''=168e^4x(e^2x+1)^5 + 28e^2x(e^2x+1)^6 (product rule)
y''=28e^2x(e^2x+1)^5(7e^2x+1)

Ok I understand everything except for the last part, how did you factorise to get 28e^2x(e^2x+1)⁵(7e^2x+1)???

 

[Slidey]

Take 28e^2x(e^2x+1) out of both terms.

Remember e^4x=e^2x*e^2x
and (e^2x+1)^6=(e^2x+1)^5(e^2x+1)

 

[Avenger6]

Take 28e^2x(e^2x+1) out of both terms.

Remember e^4x=e^2x*e^2x
and (e^2x+1)^6=(e^2x+1)^5(e^2x+1)

Ok but should the answer not be 28e^2x(e^2x+1)(6e^2x+1) because 28x6=168 and as 28x7=196???

 

[Slidey]

Ok but should the answer not be 28e^2x(e^2x+1)(6e^2x+1) because 28x6=168 and as 28x7=196???

y''=168e^4x(e^2x+1)^5 + 28e^2x(e^2x+1)^6
y''=28e^2x(e^2x+1)^5(6e^2x+e^2x+1)
y''=28e^2x(e^2x+1)^5(7e^2x+1)

 

[Avenger6]

Thank you very much, I knew there must have been something I was not seeing. Thanks again.

 

[mykelgw]

ok....

had a similar problem to the person who asked this question....

Slidey posted:
y''=168e^4x(e^2x+1)^5 + 28e^2x(e^2x+1)^6
y''=28e^2x(e^2x+1)^5(6e^2x+e^2x+1)
y''=28e^2x(e^2x+1)^5(7e^2x+1)

I was able to get the first part to this response by my self...but for some odd reason i can't seem to understand the second part, what i do understand is that he has taken out 28e^2x.

And this is where my problem starts....

He has differentiated the (e^2x+1)^6.
and got (6e^2x+e^2x+1)...but shouldn't one get: 6e^2x(e^2x+1) ?
Therefore getting: 28e^2x(e^2x+1)^5(6e^2x(e^2x+1))

Furthermore are you allowed to just differentiate one part of a problem, like has been done above??

Help is appreciated!

Sounds silly i know, but i can't get my head around it .