Question

question: use the quotient rule to find the derivative, $h(x)$, of the following function. $h(x)=\frac{2x^{2}+9x - 6}{6x}$. enter an exact answer. provide your answer below. $h(x)=square$

Explanation:

Step1: Recall quotient - rule formula

The quotient - rule states that if $h(x)=\frac{u(x)}{v(x)}$, then $h^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^2}$. Here, $u(x)=2x^{2}+9x - 6$ and $v(x)=6x$.

Step2: Find $u^{\prime}(x)$ and $v^{\prime}(x)$

Differentiate $u(x)$: $u^{\prime}(x)=\frac{d}{dx}(2x^{2}+9x - 6)=4x + 9$. Differentiate $v(x)$: $v^{\prime}(x)=\frac{d}{dx}(6x)=6$.

Step3: Apply the quotient - rule

$h^{\prime}(x)=\frac{(4x + 9)\times(6x)-(2x^{2}+9x - 6)\times6}{(6x)^{2}}$.
Expand the numerator:
\[

$$\begin{align*} &(4x + 9)\times(6x)-(2x^{2}+9x - 6)\times6\\ =&24x^{2}+54x-(12x^{2}+54x - 36)\\ =&24x^{2}+54x - 12x^{2}-54x + 36\\ =&12x^{2}+36 \end{align*}$$

\]
So, $h^{\prime}(x)=\frac{12x^{2}+36}{36x^{2}}=\frac{x^{2}+3}{3x^{2}}$.

Answer:

$\frac{x^{2}+3}{3x^{2}}$