Question

How do you differentiate `f(x)=(x^3-x)(cotx-2x)` using the product rule?

Answer 1

`(df)/(dx)=-(csc^2x+8)x^3+3cotx*x^2+(csc^2x+4)x-cotx`

Explanation:

Product rule states that if `f(x)=g(x)xxh(x)`, then

`(df)/(dx)=g(x)xx(dh)/(dx)+h(x)xx(dg)/(dx)`

Here `g(x)=(x^3-x)` and `h(x)=(cotx-2x)`

Hence `(df)/(dx)=(x^3-x)xx(-csc^2x-2)+(cotx-2x)xx(3x^2-1)` or

`(df)/(dx)=-x^3csc^2x+xcsc^2x-2x^3+2x+3x^2cotx-6x^3-cotx+2x` or

`(df)/(dx)=-(csc^2x+8)x^3+3cotx*x^2+(csc^2x+4)x-cotx`