 find f ′(a) at the given number a.
f(x) = 5x + 1,    a = 3

The given function is \( f(x) = 5x + 1 \). To find \( f'(a) \), where \( a = 3 \), we need to first compute the derivative of \( f(x) \), then substitute \( a = 3 \).

### Step 1: Compute the derivative \( f'(x) \)
Since \( f(x) = 5x + 1 \), the derivative is:
\[
f'(x) = 5
\]
This is because the derivative of \( 5x \) is 5, and the derivative of a constant (1) is 0.

### Step 2: Evaluate the derivative at \( a = 3 \)
Since \( f'(x) = 5 \), the derivative does not depend on \( x \), so:
\[
f'(3) = 5
\]

Thus, \( f'(3) = 5 \).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the derivative of a linear function in general?
2. How does the value of \( f'(x) \) change for functions like \( f(x) = ax + b \)?
3. How would you interpret \( f'(a) \) geometrically?
4. How do you find the derivative of a more complex polynomial function?
5. What happens to the derivative if the function contains higher powers of \( x \)?

**Tip:** The derivative of any constant term in a function is always zero, as constants do not change with \( x \).

find f ′(a) at the given number a. f(x) = 5x + 1, a = 3

Learn how to find the derivative of the linear function f(x) = 5x + 1 and evaluate it at x = 3. This problem explores the concept of derivatives in calculus, focusing on a linear function. Suitable for high school students in Grades 10-12.

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