Expert: Paul Klarreich - 10/9/2009

Question
QUESTION: I cant figure out these problems. They are due by midnight thursday october 8th ithink! help
1. g(u)=(1)/((15*u+6)^(1/2)) find g'(u)
2. f(x)=x^8+3*x^3+3*x find the second derivative of f''(x)
3.f(x)=(5*x+15)^4  find the second derivative of f"(x)

ANSWER: Questioner: Melina
Country: United States
Category: Calculus
Private: No
Subject: derivatives
Question: I cant figure out these problems. They are due by midnight thursday october 8th ithink! help
1. g(u)=(1)/((15*u+6)^(1/2)) find g'(u)
2. f(x)=x^8+3*x^3+3*x find the second derivative of f''(x)
3.f(x)=(5*x+15)^4  find the second derivative of f"(x)
.................................
Hi, Melina,

Since you are in a hurry, I'll get you started and you'll finish up.

1. g(u)=(1)/((15*u+6)^(1/2)) find g'(u)

g(u) = (15u + 6)^-1/2

Use the chain rule:

v = 15u + 1,  dv/du = 15

dg/dv = -1/2 v^-3/2

.....................
2. f(x)=x^8+3*x^3+3*x find the second derivative of f''(x)

Careful! Do you mean "second derivative of f''(x)" or
"second derivative which is f''(x)"

f' = 8x^7 + 6x^2 + 3

Now f'' is the derivative of that.
...................................
3. f(x)=(5x+15)^4  find the second derivative of f"(x)

Chain rule again,  f(x) =  u^4,  u = 5x+15

Then repeat the process.


---------- FOLLOW-UP ----------

QUESTION: i have one more i just cant figure out
Suppose u(t)=w(t^2+4) and w'(5)=10. Find u'(1)?

Answer
QUESTION: i have one more i just cant figure out
Suppose u(t)=w(t^2+4) and w'(5)=10. Find u'(1)?

Your notation is a little obscure (perhaps the book did it on purpose), but I'll give it a try.

Let  x = t^2 + 4.  Now  when t = 1,  x = 5

u'(t) = du/dt = du/dx dx/dt = w'(x) x'(t)   << chain rule

And  x'(t) = 2t

u'(1) =  w'(5) x'(1)

u'(1) =  10 * 2 = 20.