𝑓(𝑥)=3𝑥+3𝑥‾‾√
.

(a) Evaluate 𝑓′(9)=
 
(7)/(2)
functionsequation editor
(b) Use your answer from (a) to find the equation of the perpendicular line to the curve 𝑦=𝑓(𝑥)
 at 𝑥=9
.

𝑦

We are given the function:

\[
f(x) = 3x + 3\sqrt{x}
\]

### (a) Evaluate \( f'(9) \)

First, let's find the derivative \( f'(x) \) of the function. Using basic differentiation rules, we have:

\[
f'(x) = \frac{d}{dx}(3x) + \frac{d}{dx}(3\sqrt{x})
\]

1. The derivative of \( 3x \) is simply 3.
2. The derivative of \( 3\sqrt{x} \) can be found using the power rule. Recall that \( \sqrt{x} = x^{1/2} \), so:

\[
\frac{d}{dx}(3\sqrt{x}) = 3 \cdot \frac{1}{2}x^{-1/2} = \frac{3}{2\sqrt{x}}
\]

Thus, the derivative of the function \( f(x) \) is:

\[
f'(x) = 3 + \frac{3}{2\sqrt{x}}
\]

Now, we can evaluate \( f'(9) \):

\[
f'(9) = 3 + \frac{3}{2\sqrt{9}} = 3 + \frac{3}{2 \cdot 3} = 3 + \frac{3}{6} = 3 + \frac{1}{2} = \frac{7}{2}
\]

So, the value of \( f'(9) \) is:

\[
f'(9) = \frac{7}{2}
\]

### (b) Equation of the Perpendicular Line to the Curve at \( x = 9 \)

To find the equation of the line perpendicular to the curve at \( x = 9 \), we need to use the fact that the slope of the perpendicular line is the negative reciprocal of the slope of the tangent line at \( x = 9 \).

The slope of the tangent line at \( x = 9 \) is \( f'(9) = \frac{7}{2} \), so the slope of the perpendicular line will be:

\[
m_{\text{perpendicular}} = -\frac{1}{f'(9)} = -\frac{1}{\frac{7}{2}} = -\frac{2}{7}
\]

Next, we need the point on the curve at \( x = 9 \), which is \( (9, f(9)) \). We calculate \( f(9) \):

\[
f(9) = 3(9) + 3\sqrt{9} = 27 + 9 = 36
\]

Thus, the point of tangency is \( (9, 36) \).

Now, we can use the point-slope form of the line equation:

\[
y - y_1 = m(x - x_1)
\]

where \( (x_1, y_1) = (9, 36) \) and \( m = -\frac{2}{7} \). Substituting these values:

\[
y - 36 = -\frac{2}{7}(x - 9)
\]

Simplifying:

\[
y - 36 = -\frac{2}{7}x + \frac{18}{7}
\]

\[
y = -\frac{2}{7}x + \frac{18}{7} + 36
\]

Converting 36 into a fraction:

\[
y = -\frac{2}{7}x + \frac{18}{7} + \frac{252}{7}
\]

\[
y = -\frac{2}{7}x + \frac{270}{7}
\]

Thus, the equation of the perpendicular line is:

\[
y = -\frac{2}{7}x + \frac{270}{7}
\]

### Summary:
- \( f'(9) = \frac{7}{2} \)
- The equation of the perpendicular line is \( y = -\frac{2}{7}x + \frac{270}{7} \)

Do you want more details on any step, or have questions?

#### Related Questions:
1. How do you find the equation of the tangent line at \( x = 9 \)?
2. What is the geometric interpretation of the slope of the perpendicular line?
3. How does the power rule work when differentiating fractional exponents?
4. What are the conditions for two lines to be perpendicular?
5. Can we extend this process to find perpendicular lines in 3D geometry?

#### Tip:
Remember that the slope of a perpendicular line is always the negative reciprocal of the original line's slope!

Evaluate f'(9) for f(x) = 3x + 3√x and find the equation of the perpendicular line to the curve y = f(x) at x = 9.

Learn how to compute the derivative f'(9) for the function f(x) = 3x + 3√x and find the equation of the perpendicular line to the curve at x = 9. This solution covers key differentiation techniques and the concept of perpendicular slopes, ideal for high school students in Grades 10-12.

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