Example. Let

f(x)=(x^(2)+1)

and

g(x)=(x^(3)-3x)

. Compute:\

(d)/(dx)f(x)g(x)

\ Explanation. Write with me\

(d)/(dx)f(x)g(x)=f(x)g^(')(x)+f^(')(x)g(x)\ =(x^(2)+1)

\ We could stop here, but we should show that expanding this out recovers our previous result. Write\

(x^(2)+1)(3x^(2)-3)+2x(x^(3)-3x)\ =3x^(4)-3x^(2)+3x^(2)-3+2x^(4)-6x^(2)

\ which is precisely what we obtained before.

Example. Let f(x)=(x2+1) and g(x)=(x33x). Compute: dxdf(x)g(x) Explanation. Write with me dxdf(x)g(x)=f(x)g(x)+f(x)g(x)=(x2+1)()+()(x33x). We could stop here, but we should show that expanding this out recovers our previous result. Write (x2+1)(3x23)+2x(x33x)=3x43x2+3x23+2x46x2 which is precisely what we obtained before