Differentiate.
y= (cos x)^x
u= cos x
du= sin x dx
ln y = ln(cos x)^x
ln y = x ln(cos x)
(dy/dx)/(y)= ln(cos x)
(dy/dx)= y ln(cos x)
= (cos x)^x * (ln cos x)
(dx/du)= x(cos x)^(x1) * (sin x)
=  x sin(x)cos^(x1)(x)
(dy/dx)(dx/du)= [(cos^x(x))(ln(cos(x)))(x sin(x)cos^(x1)(x)]
(dy/du)= cos^x(x)*(ln(cos(x)))(x tan(x))
Is this correct?
Also, I am stuck on a different problem.
Differentiate.
y= arctan(arcsin(sqrt(x)))
u= arcsin(sqrt(x))
du= (1/(sqrt(1x^2))) dx
ln y = ln ?? do I put the whole original here?
 👍
 👎
 👁
 ℹ️
 🚩
2 answers

ln y = x ln(cos x) I agree.
y'/y=x/cosx + ln(cosx) which changes the rest.
check that.
I would do the next this way.
y= arctan u
y'=d(arctan u) du
now u= arcsin(z)
du= d arcsinZ dz
and z= sqrtX
dz= 1/2sqrtX dx
so do that substitution, and you are done. 👍
 👎
 ℹ️
 🚩
👤bobpursley 
I'm sorry, but I'm confused on the 2nd part?
 👍
 👎
 ℹ️
 🚩
Chelsea
Answer this Question
Related Questions

math
Find the values of sin θ, cos θ, and tan θ for the given right triangle (in the link below). Give the exact values. www.webassign.net/aufexc2/85003.gif sin θ= cos θ= tan θ= my answer is c^2 = a^2 +b^2 c^2 = 5^2+12^2 c^2 = 169 c= √(169) c= 13 I

math
Can you please check my work. A particle is moving with the given data. Find the position of the particle. a(t) = cos(t) + sin(t) s(0) = 2 v(0) = 6 a(t) = cos(t) + sin(t) v(t) = sin(t)  cos(t) + C s(t) = cos(t)  sin(t) + Cx + D 6 = v(0) = sin(0) cos(0)

Calculus
For the functions f(x) = sin x, show with the aid of the elementary formula sin^2 A = 1/2(1cos 2A) that f(x+y)  f(x) = cos x sin y2 sin x sin^2 (1/2y).

PreCalc
1) Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number. cos 12° cos 18° − sin 12° sin 18° And Find its exact value. 2) Use an Addition or Subtraction Formula to write the expression as a

Trigonometry
Prove the identity sin(x+y+z)+sin(x+yz)+sin(xy+z)+ sin(xyz) = 4 sin(x)cos(y)cos(z) This identity is so long and after i tried to expand the left side and it just looked something crap Thanks for you help :)

selfstudy calculus
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t)=cos(t)I cos(t)j+sin(t)k I don't know what to do. I let x=cos(t), y=cos(t) and z= sin(t). Should I let t be any number and get the equal

math
Express each of the following in terms of another angle between 0 degrees and 180 degrees a. sin 50 degrees b. sin 150 degrees c. cos 45 degrees d. cos 120 degrees please explain to me how to answer this question you don't have to answer it , but please

calculus
Find the points on the curve y= (cos x)/(2 + sin x) at which the tangent is horizontal. I am not sure, but would I find the derivative first: y'= [(2 + sin x)(sin x)  (cos x)(cos x)]/(2 + sin x)^2 But then I don't know what to do or if that is even

trigonometry
Given that sin(a) = 2/3 and cos(b) = −1/5, with a and b both in the interval [𝜋/2, 𝜋), find the exact values of sin(a + b) and cos(a − b).

PreCal
Write the trigonometric expression in terms of sine and cosine, and then simplify. 1). (csc θ − sin θ)/(cos θ) ____________. 2). Simplify the trigonometric expression. (cos u + 1)/(sin u) + (sin u)/(1 + cos u) ______________.

calculus
Find complete length of curve r=a sin^3(theta/3). I have gone thus (theta written as t) r^2= a^2 sin^6 t/3 and (dr/dt)^2=a^2 sin^4(t/3)cos^2(t/3) s=Int Sqrt[a^2 sin^6 t/3+a^2 sin^4(t/3)cos^2(t/3)]dt =a Int Sqrt[sin^4(t/3){(sin^2(t/3)+cos^2(t/3)}]dt=a Int

Calculus 3
Let r(t) = < sin(6t), cos(6t), sin(6t)cos(12t) >. Find the point where r(t) intersects the xyplane on the interval π/6 < t < 3/12π.

math
Without using a calculator, choose (a) the larger of cos 40 and cos 50; (b) the larger of sin 40 and sin 50. Be prepared to explain your reasoning.

Trigonometry
Choose the right expression in terms of a single trigonometric function. cos space x space cos space 2 x space plus space sin space x space sin space 2 x

Trigonometry
Find the exact solutions of the equation in the interval [0, 2π). (cos 2x)  (cos x)=0

physics
A hiker begins a trip by first walking 25.0 km 45.0° south of east from her base camp.On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.Find the magnitude of the displacement

Math
3. find the four angles that define the fourth root of z1=1+ sqrt3*i z = 2 * (1/2 + i * sqrt(3)/2) z = 2 * (cos(pi/3 + 2pi * k) + i * sin(pi/3 + 2pi * k)) z = 2 * (cos((pi/3) * (1 + 6k)) + i * sin((pi/3) * (1 + 6k))) z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 +

Math
Suppose f(x) = sin(pi*cosx) On any interval where the inverse function y = f –1(x) exists, the derivative of f –1(x) with respect to x is: a)1/(cos(pi*cosx)), where x and y are related by the equation (satisfy the equation) x=sin(pi*cosy)

Trig
Find all solutions w between 0 and 360, inclusive: (a) cosw = cos(−340) (b) cosw = sin 20 (c) sinw = cos(−10) (d) sin w < − 1/2 (e) 1 < tanw

Calculus
Find d^5/dx^5 g(x) for g(x)=sinx+5x^4 A. cos x B. sin x + 120 C. cos x +20x^3 D. cos x + 120 x E. sin x
Still need help?
You can ask a new question or browse existing questions.