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### Given:
\( f(x) = 12x^{-3} \)
\( a = 5 \)

You are asked to:
1. Compute \( f'(x) \), the derivative of \( f(x) \).
2. Find the equation of the tangent line to the graph at \( x = a = 5 \).

### Step 1: Compute \( f'(x) \)
The function is \( f(x) = 12x^{-3} \). Using the power rule for derivatives, which states:

\[
\frac{d}{dx}(x^n) = nx^{n-1}
\]

For \( f(x) = 12x^{-3} \), the derivative \( f'(x) \) is:

\[
f'(x) = 12 \cdot (-3) \cdot x^{-4} = -36x^{-4}
\]

So, \( f'(x) = -36x^{-4} \).

### Step 2: Evaluate \( f'(x) \) at \( x = 5 \)
Now, plug \( x = 5 \) into \( f'(x) \):

\[
f'(5) = -36(5)^{-4} = -36 \cdot \frac{1}{625} = -\frac{36}{625}
\]

### Step 3: Equation of the Tangent Line
The equation of the tangent line at \( x = 5 \) is given by the point-slope form:

\[
y - f(a) = f'(a)(x - a)
\]

Here:
- \( a = 5 \),
- \( f'(5) = -\frac{36}{625} \),
- \( f(5) \) is the value of the function at \( x = 5 \).

First, find \( f(5) \):

\[
f(5) = 12(5)^{-3} = 12 \cdot \frac{1}{125} = \frac{12}{125}
\]

Now, using the point-slope formula:

\[
y - \frac{12}{125} = -\frac{36}{625}(x - 5)
\]

This is the equation of the tangent line. You can leave it in this form or simplify it further if required.

### Summary:
1. \( f'(x) = -36x^{-4} \)
2. \( f'(5) = -\frac{36}{625} \)
3. The tangent line equation: \( y - \frac{12}{125} = -\frac{36}{625}(x - 5) \)

Would you like further details or help with simplifying the tangent line equation?

Here are 5 related questions for further practice:
1. How would the derivative change if the power of \( x \) were different?
2. What is the geometric interpretation of a negative derivative at a point?
3. How can we generalize the method for finding tangent lines for any polynomial function?
4. What role does the point-slope form play in finding the equation of a tangent line?
5. How would the tangent line equation change if the function was a polynomial of higher degree?

**Tip:** When working with derivatives, always check if you can simplify expressions involving powers of \( x \) before applying the rules!

Compute f'(x) and find the equation of the tangent line to the graph at x = a, where f(x) = 12x^(-3) and a = 5.

Learn how to compute the derivative of f(x) = 12x^(-3) and find the equation of the tangent line at x = 5. This problem involves applying the power rule for derivatives and using the point-slope form of a line.

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