# find f prime of a , { f(t) = \frac{2t+1}{t+3} }

## Question:

find f prime of a , {eq}f(t) = \frac{2t+1}{t+3} {/eq}

## Quotient Rule:

The quotient rule is one of the rules among the set of rules of derivatives. This rule is used to find the derivative of a rational function (where both the numerator and the denominator are the functions of a variable). It states:

$$\dfrac{d}{dx} \left( \dfrac{u}{v} \right) = \dfrac{vu'-uv'}{v^2}$$

The given function is:

$$f(t) = \dfrac{2t+1}{t+3} = \dfrac{u}{v}$$

Then we get:

\begin{align} &u=2t+1; \,\,\, v=t+3 \\ &u'=2;\,\,\, ;\,\,\,v'=1 \end{align}

Now we substitute all these values in the quotient rule which states:

$$\dfrac{d}{dx} \left( \dfrac{u}{v} \right) = \dfrac{vu'-uv'}{v^2}$$

Then we get:

\begin{align} f'(t) & = \dfrac{(t+3)(2) - (2t+1)(1)}{(t+3)^2} \\ &= \dfrac{2t+6-2t-1}{(t+3)^2}\\ & = \dfrac{5}{(t+3)^2} \end{align}

Now substitute {eq}x=a {/eq}:

$$f'(a) =\dfrac{5}{(a+3)^2}$$

Therefore: {eq}\boxed{\mathbf{f'(a) =\dfrac{5}{(a+3)^2}}} {/eq}