1. ## derivatives help

Alright couple problems I'm having a bit of trouble with.

If , then f ''(x) = ?

and

Suppose and i.e., and Also suppose That information implies
and

I'm just starting to understand getting the first derivative and so getting teh second derivative is kicking my butt. Then the second one have no clue who you would analyze that.

2. Hello, UMStudent!

If f(x) =9·ln|sec(x) + tan(x)|, . find f''(x).
I bet you didn't simplify the first derivative . . .

. . . . . . . . sec(x)tan(x) + sec²(x) . . . . . sec(x)[tan(x) + sec(x)]
f'(x) .= .9 · -------------------------- .= .9 · --------------------------- .= .9·sec(x)
. . . . . . . . . . sec(x) + tan(x) . . . . . . . . . . sec(x) + tan(x)

Then: .f''(x) .= .sec(x)tan(x)

Suppose: F(x) = f(f(x)) .and .G(x) = [f(x)]²

Also suppose: f(4) = 5, f(5) = 2, f'(5) = 9, f'(4) = 10.

Find F'(4) and G'(4).
It's a matter of taking baby-steps . . .

We have: .F(x) .= .f(f(x))

. . Then: .F'(x) .= .f'(f(x))·f'(x) . . . Chain Rule

Hence: .F'(4) .= .f'(f(4))·f'(4)
. . . . . . . . . . . . . . . . .
. . . . . . . . . .= . .f'(5) · 10
. . . . . . . . . . . . . .
. . . . . . . . . .= . . .9 · 10 . = . 90

We have: .G(x) .= .[f(x)]²

. . Then: .G'(x) .= .2·f(x)·f'(x)

Hence: .G'(4) .= .2·f(4)·f'(4)
. . . . . . . . . . . . . . . . .
. . . . . . . . . .= . 2 · 5 · 10 . = . 100