Question

Find the derivatives using quotie y=(x)/(x+1)

Answer 1

The derivative of the function y = x/(x+1) is found using the quotient rule, resulting in the derivative
`f'(x) = (1)/((x+1)^2).`

Step-by-step explanation:

To find the derivative of the function
`y = (x)/(x+1)`, we use the quotient rule which states that the derivative of a quotient of two functions is:

`f'(x) = (u'(x)v(x) - u(x)v'(x))/([v(x)]^2)`

Let u(x) = x and v(x) = x + 1. Thus, u'(x) = 1 and v'(x) = 1. Applying the quotient rule:

`f'(x) = (1\cdot(x+1) - x\cdot1)/((x+1)^2)`

Simplifying the numerator:

`f'(x) = (x+1 - x)/((x+1)^2) = (1)/((x+1)^2)`

Therefore, the derivative of
`y = (x)/(x+1)` is
`f'(x) = (1)/((x+1)^2).`