Question

What is the derivative of `sinx(tanx)`?

Answer 1

`sinx(sec^2x+1)`

Explanation:

Differentiate using the `color(blue)"product rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(f(x)=g(x)h(x)" then" f'(x)=g(x)h'(x)+h(x)g'(x))color(white)(a/a)|)))`

`g(x)=sinxrArrg'(x)=cosx`

`h(x)=tanxrArrh'(x)=sec^2x`
`"--------------------------------------------------"`
Substitute these values into f'(x)

`rArrf'(x)=sinxsec^2x+tanxcosx`

Now `tanxcosx=sinx/cancel(cosx)xxcancel(cosx)=sinx`

`rArrf'(x)=sinxsec^2x+sinx=sinx(sec^2x+1)`