We are given the function #f(x) = y = Sin x/(2x-1)# #color(red)(Expression.1)#
We need to find the First Derivative of #f(x)#
By observing #f(x)# we know that we must use the Quotient Rule to differentiate .
Quotient Rule for finding the derivatives states that
#color(blue)((dy)/(dx)[f(x)/g(x)] = [(g(x)*f'(x) - f(x)*g'(x))/[g(x)]^2]#
Using the Quotient Rule, we can write our #color(red)(Expression.1)# as
#(d/dx(sin x)(2x-1) - sin x(d/dx)(2x-1)]/(2x-1)^2# #color(red)(Expression.2)#
We know that
#color(blue)(d/dx(sin x)# is #color(green)(Cos x)# #color(red)(..1)#
We can differentiate #color(blue)(2x-1)# as follows:
#color(blue)(d/dx(2x - 1) =) # #color(green)(2d/dxx + (d/dx)(-1)# #color(red)(..2)#
We know that
#color(blue)(2d/dx(x) = 2*1)# and #color(green)(d/dx(-1) = 0)# #color(red)(..3)#
Hence,
using our intermediate results #color(red)(1,2, and 3)# and our #color(red)(Expression.2) #
we can write
#color(blue)((Cos x*(2x - 1) - 2 * Sin x)/(2x-1)^2)# #color(red)(..2)#
We can write #color(red)((2x-1)^2# as #color(red)((2x-1)(2x-1))#
We can split the terms in #color(red)(..2)# as follows and write our result:
#color(blue)((Cos x*(2x - 1))/((2x-1)(2x-1)) - (2*sin x)/((2x-1)(2x-1)#
We can now cancel common factors to simplify:
#color(blue)((Cos x*cancel(2x - 1))/(cancel(2x-1)(2x-1)) - (2*sin x)/((2x-1)(2x-1)#
#color(blue)(cos x/(2x-1) - (2*sinx)/((2x-1)^2))#
I hope this helps you to understand how the Quotient Rule for differentiation works.
