Question

Differentiate the following function? y = (8 x)/ (6 - cot x)

Answer 1

`dy/dx = 8( (6-cotx - xcsc^2x) ) / (6-cotx)^2`

Explanation:

We apply the Quotient Rule for Differentiation:

`d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2`, or less formally, `" "(u/v)' = (v(du)-u(dv))/v^2`

I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So with `y=(8x)/(6-cotx)` Then

`{ ("Let "u=8x, => , (du)/dx=8), ("And "v=6-cotx, =>, (dv)/dx=csc^2x ) :}`

`:. dy/dx = (v(du)/dx-u(dv)/dx)/v^2`
`" " = ( (6-cotx)(8) - (8x)(csc^2x) ) / (6-cotx)^2`
`" " = 8( (6-cotx - xcsc^2x) ) / (6-cotx)^2`