Question

How do you differentiate `f(x)=xsinx+cosx`?

Answer 1

`f'(x)=xcosx`

Explanation:

`color(orange)"Reminder"`

`• d/dx(sinx)=cosx" and " d/dx(cosx)=-sinx`

`"to differentiate "xsinx" use the "color(blue)"product rule"`

`"Given "f(x)=g(x)h(x)" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=g(x)h'(x)+h(x)g'(x))color(white)(2/2)|)))larr" product rule"`

`"here "g(x)=xrArrg'(x)=1`

`"and "h(x)=sinxrArrh'(x)=cosx`

`rArrf'(x)=xcosx+sinxlarr" derivative of "xsinx`

Differentiating the original f(x)

`f(x)=xsinx+cosx`

`rArrf'(x)=xcosx+sinx-sinx`

`rArrf'(x)=xcosx`