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We are given the function \( f(x) = 5x^2 + 10x - 3 \), and we have already computed its derivative as \( f'(x) = 10x + 10 \).

The final step is to find the equation of the tangent line to \( f(x) \) at \( x = 2 \).

### Steps to find the tangent line:
1. **Find the slope of the tangent line**: The slope of the tangent line is given by \( f'(x) \). At \( x = 2 \), we calculate:
   \[
   f'(2) = 10(2) + 10 = 20 + 10 = 30
   \]
   So, the slope of the tangent line is 30.

2. **Find the point on the curve**: We need to find the value of \( f(2) \) to get the coordinates of the point where the tangent touches the curve. Calculate:
   \[
   f(2) = 5(2)^2 + 10(2) - 3 = 5(4) + 20 - 3 = 20 + 20 - 3 = 37
   \]
   So, the point of tangency is \( (2, 37) \).

3. **Use the point-slope form of the equation of a line**: The point-slope form 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. Substituting \( m = 30 \), \( x_1 = 2 \), and \( y_1 = 37 \):
   \[
   y - 37 = 30(x - 2)
   \]
   Expanding this:
   \[
   y - 37 = 30x - 60
   \]
   \[
   y = 30x - 60 + 37
   \]
   \[
   y = 30x - 23
   \]

So, the equation of the tangent line at \( x = 2 \) is:
\[
y = 30x - 23
\]

Would you like further details or explanations on any part of this process? Here are five related questions for further exploration:

1. How do we use limits to derive the derivative of a function?
2. Why is the slope of the tangent line equal to the derivative at a point?
3. What is the geometric interpretation of the derivative in terms of tangent lines?
4. How would the equation of the tangent line change if \( x = 3 \) instead of \( x = 2 \)?
5. How can we find the normal line to the curve at \( x = 2 \)?

### Tip:
When finding the tangent line to a curve, always ensure you compute both the slope (using the derivative) and the point of tangency (by evaluating the function at the given point).

Let f(x) = 5x^2 + 10x - 3. Find the equation of the tangent line to f(x) where x = 2.

Learn how to find the equation of the tangent line to the curve f(x) = 5x^2 + 10x - 3 at x = 2. This problem involves calculating the derivative of the function and using the point-slope form to find the tangent line. Suitable for students in Grades 11-12.

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