Question

How do you differentiate `f(x)=xcscx` using the product rule?

Answer 1

`f'(x)=cscx (1- xcotx)`

Explanation:

Note: `F(x) = f(x)*g(x)`
Then derivative : `F'(x) = f'(x)*g(x) + g'(x)*f(x)`

Note: `f(x) = csc x hArr f'(x)= -csc x cot x`

==================

given `f(x) = x csc x`
`f'(x) = d/dx (x) * csc x + x d/dx cscx`

`= 1* csc x + x*(-csc x cotx )`
`= csc x - x cscx cot x`
`color(red)(=cscx (1- xcotx)`

========================
FYI: How to derive derivative of `f(x) = csc x`
`f(x) = 1/(sinx)`

(using quotient rule)

`f'(x) = (-cos x)/(sin^2x)`

`f'(x) = (1/sinx)*(-cos x/sinx)`

`f'(x)= -csc x cot x`