How do you differentiate f(x)= (2 x^2 + 7 x - 2)/ (x - cos x ) using the quotient rule?

1 Answer
Jul 14, 2017

d/(dx) [(2x^2+7x-2)/(x-cosx)] = color(blue)((4x+7)/(x-cosx) - ((2x^2 + 7x - 2)(1+sinx))/((x-cosx)^2)

Explanation:

We're asked to find the derivative

d/(dx) [(2x^2 + 7x - 2)/(x-cosx)]

Use the quotient rule, which is

d/(dx) [u/v] = (v(du)/(dx) - u(dv)/(dx))/(v^2)

where

  • u = 2x^2 + 7x - 2

  • v = x - cosx:

= ((x-cosx)(d/(dx)[2x^2 + 7x - 2])-(2x^2 + 7x - 2)(d/(dx)[x-cosx]))/((x-cosx)^2)

The derivative of 2x^2 + 7x - 2 is 4x + 7 (use power rule for each term):

= ((x-cosx)(4x+7)-(2x^2 + 7x - 2)(d/(dx)[x-cosx]))/((x-cosx)^2)

The derivative of x is 1 (power rule) and the derivative of cosx is -sinx:

= color(blue)(((x-cosx)(4x+7)-(2x^2 + 7x - 2)(1+sinx))/((x-cosx)^2)

Which can also be written as

= color(blue)((4x+7)/(x-cosx) - ((2x^2 + 7x - 2)(1+sinx))/((x-cosx)^2)