Question

Find the first derivative of the following functions using the quotient rule.

Answer 1

Given:

`f(x)=(x)/(x+1)`

To find: The first derivative

Step-by-step explanation:

Using the quotient rule,

`((u)/(v))^(\prime)=(vu^(\prime)-uv^(\prime))/(v^2)`

Here,

`\begin{gathered} u=x \\ u^(\prime)=(d)/(dx)(x) \\ \Rightarrow u^(\prime)=1 \\ v=x+1 \\ v^(\prime)=(d)/(dx)(x+1) \\ \Rightarrow v^(\prime)=1 \end{gathered}`

On substitution we get,

`\begin{gathered} f^(\prime)(x)=((x+1)(1)-x(1))/((x+1)^2) \\ f^(\prime)(x)=(x+1-x)/((x+1)^2) \\ f^(\prime)(x)=(1)/((x+1)^2) \end{gathered}`

Final answer: The first derivative of the given function is,

`f^(\prime)(x)=(1)/((x+1)^(2))`