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The image provided contains the formula for the slope of the tangent line to the graph of a function \( f(x) \) at a point \( x = a \), given by:

\[
m_{\text{tan}} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]

You are asked to calculate the slope of the tangent line to the function \( f(x) = 5x^2 + 3 \) at the point \( x = 1 \).

### Step-by-step process:

1. **Identify the function**:
   \[
   f(x) = 5x^2 + 3
   \]
2. **Substitute \( f(x) \) into the derivative formula** to find the slope at \( x = 1 \):

   \[
   m_{\text{tan}} = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}
   \]

3. **Compute \( f(1) \)**:
   \[
   f(1) = 5(1)^2 + 3 = 5 + 3 = 8
   \]

4. **Compute \( f(1 + h) \)**:
   \[
   f(1 + h) = 5(1 + h)^2 + 3 = 5(1 + 2h + h^2) + 3 = 5 + 10h + 5h^2 + 3 = 8 + 10h + 5h^2
   \]

5. **Substitute into the limit**:

   \[
   m_{\text{tan}} = \lim_{h \to 0} \frac{(8 + 10h + 5h^2) - 8}{h} = \lim_{h \to 0} \frac{10h + 5h^2}{h}
   \]

6. **Simplify the expression**:

   \[
   m_{\text{tan}} = \lim_{h \to 0} (10 + 5h) = 10
   \]

### Final Answer:
The slope of the tangent line at \( x = 1 \) is \( 10 \).

Would you like more details, or do you have any questions?

### 5 Related Questions:
1. What is the geometric interpretation of the slope of the tangent line?
2. How would the result change if the function were \( f(x) = 5x^2 - 3 \)?
3. Can you explain how the derivative formula comes from the definition of a limit?
4. What would the derivative of \( f(x) = 5x^2 + 3 \) be for any arbitrary value of \( x \)?
5. How do you apply this method to find the equation of the tangent line at a point?

### Tip:
The slope of the tangent line at a given point is equivalent to the instantaneous rate of change of the function at that point.

Calculate the slope of the tangent line to the graph of the function f(x) = 5x^2 + 3 at x = 1.

Learn how to calculate the slope of the tangent line to the function f(x) = 5x^2 + 3 at the point x = 1 using the definition of the derivative. This problem involves key calculus concepts like limits and derivatives. Suitable for students in Grades 11-12.

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