SOLUTION: f(t)= √t^2+2ln(t) Find the derivative of each function. Can someone help me figuring this equation out. I would really appreciate the help. Thank you.

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: f(t)= √t^2+2ln(t) Find the derivative of each function. Can someone help me figuring this equation out. I would really appreciate the help. Thank you.      Log On


   

Question 899462: f(t)= √t^2+2ln(t)
Find the derivative of each function.
Can someone help me figuring this equation out. I would really appreciate the help. Thank you.
Answer by swincher4391(1107) About Me  (Show Source):
You can put this solution on YOUR website!
Assuming the problem is sqrt%28t%5E2+%2B+2ln%28t%29%29, we know that we are going to have to do the chain rule. I find it easiest to use simple substitution to do this. You'll see what I mean in a second.
Let's let u = t^2 + 2ln(t) so du = 2t + 2/t dt [but dt is just 1 since derivative of t = 1]
Let's let v = sqrt(u) so dv = 1/2sqrt(u) * du
Going from dv we have
dv+=+1%2F2sqrt%28u%29+%2A+du
Substitute du = 2t + 2/t
1%2F%282%2Asqrt%28u%29%29+%2A+%282t+%2B+2%2Ft%29
Substituting u in terms of t, we get
1%2F%282%2Asqrt%28t%5E2%2B2ln%28t%29%29%29+%2A+%282t+%2B+2%2Ft%29
Getting rid of 2.
%28t%2B1%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29 <--- perfect acceptable
Getting the Wolfram-Alpha answer:
Get a common denominator in the numerator.
%28%28t%5E2%2B1%29%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29
Drop the t into the denominator
%28t%5E2%2B1%29%2F%28t%2Asqrt%28t%5E2%2B2ln%28t%29%29%29
So, to recap, any time you have functions within functions, we just assign them variables.
Like sin(cos(x)))
We let u = cos(x) so du = -sin(x)
Now we let v = sin(u) so dv = cos(u) du
We go from dv and get cos(u) * (-sin(x))
Substitute u back in and get cos(cos(x)) * -sin(x).
Clean it up a bit and get -cos(cos(x))*sin(x).
Hope this makes sense and makes the chain rule a little easier! Feel free to email me if it does not.