Question

question use the quotient rule to find the derivative, h(x), of the following function. h(x) = (2x^2 + 9x - 6)/(6x) enter an exact answer.

Explanation:

Step1: Recall quotient - rule

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

Step2: Find $f^{\prime}(x)$ and $g^{\prime}(x)$

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

Step3: Apply 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:

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