Question

How to differentiating using quotient rule?

How to differentiate `x/(x+1)`

Answer 1

`\frac{1}{(x+1)^2}` for all values of `x` except for `x=-1`

Explanation:

If a function, `f(x)`, can be written as

`f(x) = \frac{g(x)}{h(x)}`

where `h(x)!=0`, then the Quotient rule states that the derivative of `g(x)/(h(x))` is

`f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}` ............(1)
In the problem `f(x) = \frac{g(x)}{h(x)}=x/(x+1)`

Inserting appropriate functions in (1)
`f'(x) = \frac{(x+1)d/dx(x) - xd/dx(x+1)}{(x+1)^2}`

`f'(x) = \frac{(x+1) - x}{(x+1)^2}`
`f'(x) = \frac{1}{(x+1)^2}` for all values of `x` except for `x=-1`