Question

How do you differentiate `g(x) = (1 + sin^2(2-x))(1 +cos^2(x))` using the product rule?

Answer 1

`(dg)/(dx)=-sin(4-2x)(1+cos^2x)+sin2x(1+sin^2(2-x))`

Explanation:

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

then `(dg)/(dx)=(df)/(dx)xxh(x)+(dh)/(dx)xxf(x)`

Hence as `g(x)=(1+sin^2(2-x))(1+cos^2x)`

`(dg)/(dx)=(-2sin(2-x)cos(2-x))(1+cos^2x)+(1+sin^2(2-x))(-2cosx xx(-sinx))`

= `-sin(4-2x)(1+cos^2x)+(1+sin^2(2-x))(sin2x)`

= `-sin(4-2x)(1+cos^2x)+sin2x(1+sin^2(2-x))`