find an equation of the tangent line to $$ f\left(x\right)\text{ at }a=7 $$

To find the equation of the tangent line to the function $ f(x) $ at $ a = 7 $, we follow these steps:

### Step 1: Find the value of the function at $ x = 7 $
The tangent line passes through the point $ (7, f(7)) $. So, we first calculate $ f(7) $.

### Step 2: Find the derivative of the function, $ f'(x) $
The slope of the tangent line is given by the derivative of the function evaluated at $ x = 7 $, i.e., $ f'(7) $.

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of the equation of a tangent line is:
\[
y - f(7) = f'(7)(x - 7)
\]
Where $ (7, f(7)) $ is the point of tangency, and $ f'(7) $ is the slope.

Could you provide the explicit form of $ f(x) $ so I can calculate these values for you?

Would you like details on this process?

Here are some related questions:
1. What is the derivative of a polynomial function?
2. How do you apply the chain rule to find derivatives?
3. What is the geometric interpretation of a tangent line?
4. How can a tangent line be used to approximate function values?
5. How does the derivative at a point relate to the slope of the tangent line?

**Tip:** The derivative at a point gives the exact slope of the tangent line, which is why it's so useful in calculus!

Discover how to find the equation of the tangent line to a function f(x) at x = 7 by using calculus concepts. This detailed guide covers the derivative, the point-slope form of the tangent line, and more. Suitable for high school students in Grades 11-12 studying calculus.

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