Find an equation of the tangent line to the graph of f(x) = 1/√ 3x at x = 9

To find the equation of the tangent line to the graph of the function \( f(x) = \frac{1}{\sqrt{3x}} \) at \( x = 9 \), we will follow these steps:

### Step 1: Find the derivative of \( f(x) \)
The derivative \( f'(x) \) represents the slope of the tangent line. 

Given \( f(x) = \frac{1}{\sqrt{3x}} = (3x)^{-1/2} \), we can use the chain rule to differentiate it.

\[
f'(x) = \frac{d}{dx} \left( (3x)^{-1/2} \right)
\]
First, apply the power rule and the chain rule:
\[
f'(x) = -\frac{1}{2}(3x)^{-3/2} \cdot 3
\]
Simplifying the expression:
\[
f'(x) = -\frac{3}{2} (3x)^{-3/2}
\]

### Step 2: Evaluate the derivative at \( x = 9 \)
Now, substitute \( x = 9 \) into \( f'(x) \) to find the slope of the tangent line at this point.

\[
f'(9) = -\frac{3}{2} (3 \cdot 9)^{-3/2} = -\frac{3}{2} (27)^{-3/2}
\]

Now, simplify \( (27)^{-3/2} \). First, compute \( 27^{3/2} \):
\[
27^{1/2} = 3 \quad \text{so} \quad 27^{3/2} = 3^3 = 27
\]
Thus, \( (27)^{-3/2} = \frac{1}{27} \).

Now substitute this back into the expression for \( f'(9) \):
\[
f'(9) = -\frac{3}{2} \cdot \frac{1}{27} = -\frac{1}{18}
\]

The slope of the tangent line at \( x = 9 \) is \( m = -\frac{1}{18} \).

### Step 3: Find the y-coordinate at \( x = 9 \)
Next, find \( f(9) \) to determine the point where the tangent line touches the graph.

\[
f(9) = \frac{1}{\sqrt{3 \cdot 9}} = \frac{1}{\sqrt{27}} = \frac{1}{3\sqrt{3}}
\]

### Step 4: Use the point-slope form of the line
Now that we have the slope \( m = -\frac{1}{18} \) and the point \( (9, \frac{1}{3\sqrt{3}}) \), we can use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{1}{18} \), \( x_1 = 9 \), and \( y_1 = \frac{1}{3\sqrt{3}} \):

\[
y - \frac{1}{3\sqrt{3}} = -\frac{1}{18}(x - 9)
\]

### Final equation of the tangent line:
\[
y = -\frac{1}{18}(x - 9) + \frac{1}{3\sqrt{3}}
\]

This is the equation of the tangent line to the graph of \( f(x) = \frac{1}{\sqrt{3x}} \) at \( x = 9 \).

Would you like to go over the steps in more detail or have any specific questions?

### Related questions:
1. How do you apply the chain rule in calculus?
2. What is the power rule, and how is it used in differentiation?
3. How can we simplify expressions with negative exponents?
4. What is the geometric meaning of the derivative at a point on a curve?
5. How does the point-slope form of a line work?

### Tip:
Always ensure to simplify expressions involving roots and exponents carefully, especially when dealing with fractional powers.

Find an equation of the tangent line to the graph of f(x) = 1/√(3x) at x = 9.

Learn how to find the tangent line to the function f(x) = 1/√(3x) at x = 9 using calculus. This solution includes step-by-step differentiation using the chain rule, and application of the point-slope form of a line. Suitable for advanced high school students in Grades 11-12.

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