use your calculator to find the gradient of the graph of $y = \frac{3x - 7}{\sqrt{11 - 3x}}$ at $x = 3$.
give your answer as a decimal, rounded to 3 significant figures.

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Accepted Answer

# Explanation:
## Step1: Use the quotient rule
The quotient rule states that if $y = \frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Let $u=\sqrt{11 - 3x}=(11 - 3x)^{\frac{1}{2}}$ and $v = 3x-7$.
First, find $u^\prime$ and $v^\prime$.
Using the chain rule, if $u=(11 - 3x)^{\frac{1}{2}}$, then $u^\prime=\frac{1}{2}(11 - 3x)^{-\frac{1}{2}}\times(- 3)=-\frac{3}{2\sqrt{11 - 3x}}$.
And $v^\prime = 3$.
## Step2: Apply the quotient rule
$$
\begin{align*}
y^\prime&=\frac{-\frac{3}{2\sqrt{11 - 3x}}\times(3x - 7)-\sqrt{11 - 3x}\times3}{(3x - 7)^{2}}\
\end{align*}
$$
Substitute $x = 3$ into the derivative.
When $x = 3$, $u=\sqrt{11-3\times3}=\sqrt{2}$, $v=3\times3 - 7 = 2$, $u^\prime=-\frac{3}{2\sqrt{2}}$, $v^\prime = 3$.
$$
\begin{align*}
y^\prime&=\frac{-\frac{3}{2\sqrt{2}}\times2-\sqrt{2}\times3}{2^{2}}\
&=\frac{-3\sqrt{2}-3\sqrt{2}}{4}\
&=\frac{-6\sqrt{2}}{4}\
&=-\frac{3\sqrt{2}}{2}\approx - 2.12
\end{align*}
$$

# Answer:
$-2.12$

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Question

Explanation:

Step1: Use the quotient rule

Step2: Apply the quotient rule

Answer: