Can somebody help me w/these 2 calc/ap. calc prob.s?

Question

I need to find the derivative of these two problems:

1. sec^3(4x)

I got: 12(sec4x)^2[(sex4x)(tanx)]


2. ln(secx+tanx)

I got: [(secxtanx)+sec^2x]/(secx+tanx...


Are they right? If not, can u plz show me the steps to the correct answers?

Answer 1

1.) sec^3(4x)
Set this up with the chain rule:
g(x) = sec (4x)
f(x) = x^3

(f*g)' = f'(g(x)) * g'(x)
f'(x) = 3x^2
f'(g(x)) = 3 sec^2 (4x)

g'(x) = 4 tan 4x sec 4x

f'(g(x)) * g'(x) = 3 sec^2 (4x) * 4 tan 4x sec 4x

Why:
For the first part of the chain rule (f'(g(x))), you don't do anything with the sec (4x), you just worry about the x^3. For the second part (g'(x)), you have to pull out the derivative of 4x, but:

du/dx sec u = d/du sec u tan u

so you have to keep the 4x inside both the sec and tan portion.

2.) ln(sec x + tan x)
Again, chain rule:
(f*g)' = f'(g(x)) * g'(x)
f(x) = ln x
g(x) = sec x + tan x

f'(x) = 1/x
f'(g(x)) = 1 / (sec x + tan x)

g'(x) = sec x tan x + sec2 x

f'(g(x)) * g'(x) = 1 / (sec x + tan x) * (sec x tan x + sec2 x)
= (sec x tan x + sec2 x) / (sec x + tan x) (solution - you're correct!)

Answer 2

I not in any respect took the stats classification, yet Calculus II is undemanding. Please, do no longer concern it! that is undemanding as long as you keep in ideas calculus I ideas and guidelines. that is undemanding! I made A's in calculus a million-3 at a respectable college.