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Let's solve the problem step by step.

You are given the function:

\[
f(x) = 6x + 8x^2
\]

### Part (a): Use the limit definition of the derivative to find \( f'(x) \).

The limit definition of the derivative is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

1. Calculate \( f(x+h) \):

\[
f(x+h) = 6(x+h) + 8(x+h)^2
\]

Expanding this expression:

\[
f(x+h) = 6x + 6h + 8(x^2 + 2xh + h^2)
\]

\[
f(x+h) = 6x + 6h + 8x^2 + 16xh + 8h^2
\]

2. Subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (6x + 6h + 8x^2 + 16xh + 8h^2) - (6x + 8x^2)
\]

Simplifying:

\[
f(x+h) - f(x) = 6h + 16xh + 8h^2
\]

3. Now, divide by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{6h + 16xh + 8h^2}{h} = 6 + 16x + 8h
\]

4. Finally, take the limit as \( h \to 0 \):

\[
f'(x) = 6 + 16x
\]

### Part (b): Find the equation of the tangent line at \( x = 5 \).

To find the equation of the tangent line, use the point-slope form of a line:

\[
y - f(x_0) = f'(x_0)(x - x_0)
\]

Here, \( x_0 = 5 \), so we need to find \( f(5) \) and \( f'(5) \).

1. Calculate \( f'(5) \):

\[
f'(5) = 6 + 16(5) = 6 + 80 = 86
\]

2. Calculate \( f(5) \):

\[
f(5) = 6(5) + 8(5^2) = 30 + 8(25) = 30 + 200 = 230
\]

3. Now, use the point-slope form:

\[
y - 230 = 86(x - 5)
\]

Simplifying this equation:

\[
y = 86x - 430 + 230
\]

\[
y = 86x - 200
\]

So, the equation of the tangent line at \( x = 5 \) is:

\[
y = 86x - 200
\]

---

Would you like further explanation on any of the steps? Here are some related questions to expand your understanding:

1. How can you find the derivative of a product of two functions using the product rule?
2. What does the derivative of a function tell you about its graph?
3. How do you find the slope of the tangent line at any given point?
4. What is the geometric meaning of the second derivative of a function?
5. How would the tangent line change if the function was cubic instead of quadratic?

**Tip:** The derivative gives the rate of change of the function at any point. For polynomials, the derivative decreases the degree by one.

Use the function f(x) = 6x + 8x^2 to find the following. (a) Use the limit definition of the derivative to find f'(x). (b) Find the equation of the tangent line at x=5.

Learn how to find the derivative of f(x) = 6x + 8x^2 using the limit definition and calculate the equation of the tangent line at x=5. This problem involves core calculus concepts such as derivatives and tangent lines, suitable for Grades 11-12 or early college students.

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