Question

How do you differentiate `f(x)= ( x + 1 )/ ( x - 4csc x )` using the quotient rule?

Answer 1

`(df)/(dx)=(4xcscxcotx+4cscxcotx-4cscx-1)/(x-4cscx)^2`

Explanation:

According to quotient rule

`d/dx((f(x))/(g(x)))=(d/dx(f(x))xxg(x)-f(x)xxd/dx(g(x)))/((g(x))^2`

Hence as `f(x)=(x+1)/(x-4cscx)`

`(df)/(dx)=(d/dx(x+1)xx(x-4cscx)-(x+1)xxd/dx(x-4cscx))/(x-4cscx)^2` or
`(df)/(dx)=(1xx(x-4cscx)-(x+1)xx(1-4(-cscxcotx)))/(x-4cscx)^2` or
`(df)/(dx)=(x-4cscx-x-1+4(x+1)cscxcotx)/(x-4cscx)^2` or
`(df)/(dx)=(x-4cscx-x-1+4xcscxcotx+4cscxcotx)/(x-4cscx)^2` or
`(df)/(dx)=(4xcscxcotx+4cscxcotx-4cscx-1)/(x-4cscx)^2`