In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. This rule is given by the following expression:
[tex](\frac{f}{g})^{\prime}=\frac{f^{\prime}g-fg^{\prime}}{g^2}[/tex]
Applying this rule in our problem, we have:
[tex]\begin{gathered} f^{\prime}(x)=\frac{(x^2)^{\prime}(2\sqrt{x}+1)-(x^2)(2\sqrt{x}+1)^{\prime}}{(2\sqrt{x}+1)^2} \\ \\ =\frac{(2x)(2\sqrt{x}+1)-(x^2)(2\cdot\frac{1}{2}\frac{1}{\sqrt{x}})}{4x+4\sqrt{x}+1} \\ \\ =\frac{4x\sqrt{x}+2x-x\sqrt{x}}{4x+4\sqrt{x}+1} \\ \\ =\frac{3x\sqrt{x}+2x}{4x+4\sqrt{x}+1} \end{gathered}[/tex]
