compute f' and find an equation of the tangent line to the graph at x = a.
f(x) = 11x^-3, a=5. Use symbolic notation and fractions where needed, Let y = f(x) and express equation in terms of x and y.

f'(x) = ?

Given the function \( f(x) = 11x^{-3} \), we will first compute the derivative \( f'(x) \), then find the equation of the tangent line at \( x = a = 5 \).

### Step 1: Compute \( f'(x) \)
The function is \( f(x) = 11x^{-3} \). Using the power rule for differentiation, which states that for \( x^n \), the derivative is \( n x^{n-1} \), we differentiate:

\[
f'(x) = \frac{d}{dx} \left( 11x^{-3} \right) = 11 \cdot (-3) x^{-4} = -33x^{-4}
\]

So, the derivative of the function is:
\[
f'(x) = -\frac{33}{x^4}
\]

### Step 2: Compute \( f'(5) \)
Now, substitute \( x = 5 \) into the derivative to find the slope of the tangent line at that point:

\[
f'(5) = -\frac{33}{5^4} = -\frac{33}{625}
\]

### Step 3: Find the point on the curve at \( x = 5 \)
Next, we find \( f(5) \) to get the coordinates of the point where the tangent line touches the curve:

\[
f(5) = 11 \cdot 5^{-3} = \frac{11}{5^3} = \frac{11}{125}
\]

So the point on the graph is \( (5, \frac{11}{125}) \).

### Step 4: Write the equation of the tangent line
The equation of a line is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Here, \( m = -\frac{33}{625} \), \( x_1 = 5 \), and \( y_1 = \frac{11}{125} \).

Substitute these values into the equation of the tangent line:

\[
y - \frac{11}{125} = -\frac{33}{625}(x - 5)
\]

This is the equation of the tangent line in symbolic form.

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Would you like more details or have any questions?

### Related Questions:
1. How does the power rule apply to negative exponents?
2. Can you explain how to find the slope of a tangent line?
3. What is the geometric interpretation of the tangent line?
4. How would the equation of the tangent line change for a different value of \( a \)?
5. How does the equation of the tangent line relate to the derivative?

### Tip:
Always double-check units and signs in the derivative when solving for a tangent line, especially when dealing with negative exponents or fractions.

Compute f' and find an equation of the tangent line to the graph at x = a. f(x) = 11x^-3, a = 5. Use symbolic notation and fractions where needed, Let y = f(x) and express equation in terms of x and y.

Learn how to find the derivative of f(x) = 11x^-3 using the power rule and compute the equation of the tangent line at x = 5. This calculus problem involves differentiation and understanding how to use the slope to write the tangent line equation. Suitable for high school or college calculus students.

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