VERY URGENT calculus help!!?

Question

I asked yesterday, but the answer still doesn't match. So I'm hoping there is a math whiz out there that will be able to help!!

I tried the calculus websites and calculator, but I have no clue on how to get the answer.

The questions is to find the derivative for y=(cosx)^x, where -pi/2 < x < pi/2.

The answer is (cosx)^(x-1) (cosx ln(cosx) - xsin x))
Please help! and thank you.

If done normally, the answer is dy=(cosx)^x[ln(cosx)-xtanx]
However, I'm guessing that the restriction on x, which is (-pi/2 < x < pi/2) is what makes the answer different. Much help appreciated to the person who can figure it out! ^^

Answer 1

You usually do this type of problem by logarithmic
differentiation:
Let y = (cos x)^x,
Now take the logarithm of both sides:
ln y = x *ln(cos x).
Now you can see why x is restricted to the values given:
It's to keep the argument of the ln function positive.
The cosine is positive in this range.
Let's find your derivative:
By the chain rule, the derivative of the LHS is
y'/y, so
y'/y = x *(-sin x/cos x) + ln(cos x) (*)
and if you combine the fractions, you get
y'/y = (cos x ln(cos x) - x sin x)/cos x.
Now multiply through by y, replace y by (cos x)^x
and cancel a cos x and you get the given answer.
Your answer is also correct, but x still
must be restricted to the given range to
keep the argument of ln(cos x) positive.
To get your answer, just go back to (*),
replace sin x/cos x by tan x, then multiply
by y = (cos x)^x and you will get your answer.
Both answers are equally correct!!
If you need more calculus help you can
e-mail me at [e-mail] .

Answer 2

there isn't any favor for brackets round unmarried exponents as you've right here. there is also no favor to modify the letter used for the consistent as one answerer has carried out, because the consistent is arbitrary it could be changed in any respect you opt for. discover the final answer by technique of placing apart the variables then integrating: y' = 3t²y² dy / dt = 3t²y² dy / y² = 3t² dt ? a million / y² dy = ? 3t² dt -a million / y = t³ + C a million / y = C - t³ y = a million / (C - t³)