How do you find the second derivative of #sin(x)/(1-cos(x))#?

Question

Cameron Alexander · Accepted Answer

# d^2/dx^2(sin x/(1-cos x)) = sin x/(cos x - 1)^2 # # d/dx(sin x/(1-cos x)) = ((1-cos x)cos x - sin^2 x)/(1-cos x)^2 = 1/(cos x-1) # # d^2/dx^2(sin x/(1-cos x)) = sin x/(cos x - 1)^2 #

Paisley Johnson · Answer

You take the first derivative twice. #color(green)(d^2/(dx^2)[f(x)] = d/(dx)[d/(dx)[f(x)]])# #= d/(dx)[((1-cosx)(cosx) - sinx(0+sinx))/(1-cosx)^2]# #= d/(dx)[(cosx-cos^2x - sin^2x)/(1-cosx)^2]# #= d/(dx)[(cosx-(cos^2x + sin^2x))/(1-cosx)^2]# #= d/(dx)[(-cancel((1-cosx)))/(1-cosx)^cancel(2)]# #= d/(dx)[-1/(1-cosx)] = d/(dx)[-(1-cosx)^-1]# #= (1-cosx)^(-2)*sinx# #= color(blue)(sinx/(1-cosx)^2)#

Axel Alvarez · Answer

undefined To find the second derivative of sin(x)/(1-cos(x)), follow these steps:

1. Differentiate the function with respect to x to find the first derivative.
2. Differentiate the first derivative obtained in step 1 with respect to x to find the second derivative.

Let's denote the function as y = sin(x)/(1-cos(x)).

1. Find the first derivative (dy/dx) using the quotient rule:
   dy/dx = [(1-cos(x))(cos(x)) - sin(x)(sin(x))]/(1-cos(x))^2

2. Simplify the expression:
   dy/dx = (cos(x) - cos^2(x) - sin^2(x))/(1-cos(x))^2
   dy/dx = (cos(x) - 1)/(1-cos(x))^2

3. Now, differentiate dy/dx with respect to x to find the second derivative:
   d^2y/dx^2 = d/dx[(cos(x) - 1)/(1-cos(x))^2]

4. Apply the quotient rule again:
   d^2y/dx^2 = [(1-cos(x))^2(-sin(x)) - (cos(x) - 1)(2(1-cos(x))sin(x))]/(1-cos(x))^4

5. Simplify the expression:
   d^2y/dx^2 = [-(1-cos(x))^2sin(x) - 2(cos(x) - 1)(1-cos(x))sin(x)]/(1-cos(x))^4
   d^2y/dx^2 = [-sin(x)(1-cos(x))^2 - 2sin(x)(cos(x) - 1)(1-cos(x))]/(1-cos(x))^4

6. Further simplify the expression:
   d^2y/dx^2 = [-sin(x)(1-cos(x))^2 - 2sin(x)(1-cos(x))^2 + 2sin(x)(1-cos(x))^3]/(1-cos(x))^4
   d^2y/dx^2 = [-3sin(x)(1-cos(x))^2 + 2sin(x)(1-cos(x))^3]/(1-cos(x))^4

Therefore, the second derivative of sin(x)/(1-cos(x)) is:
d^2y/dx^2 = [-3sin(x)(1-cos(x))^2 + 2sin(x)(1-cos(x))^3]/(1-cos(x))^4