https://file.aiquickdraw.com/mathbot/file/1725850430955upn5zv5s.jpg

The problem asks to find the equation of the tangent line to the function at a specified point. From the image, the function is:

\[
y = 3x^4 - 6x \left(\frac{7}{x}\right)
\]

and the point is \( (1, 9) \).

### Steps:

1. **Simplify the given function**:  
   The function can be simplified first. Since \( x \) in the denominator cancels with the \( x \) outside, we get:
   
   \[
   y = 3x^4 - 6 \times 7
   \]
   which simplifies to:
   \[
   y = 3x^4 - 42
   \]

2. **Find the derivative (slope of the tangent line)**:  
   To find the slope of the tangent line at any point, we take the derivative of \( y \) with respect to \( x \):
   
   \[
   \frac{dy}{dx} = \frac{d}{dx} \left( 3x^4 - 42 \right)
   \]
   This results in:
   \[
   \frac{dy}{dx} = 12x^3
   \]

3. **Evaluate the derivative at \( x = 1 \)**:  
   The slope of the tangent line at \( x = 1 \) is:
   
   \[
   \frac{dy}{dx} \bigg|_{x=1} = 12(1)^3 = 12
   \]

4. **Find the equation of the tangent line**:  
   The equation of a line 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 \( m = 12 \), \( x_1 = 1 \), and \( y_1 = 9 \):
   
   \[
   y - 9 = 12(x - 1)
   \]
   Simplifying:
   \[
   y - 9 = 12x - 12
   \]
   \[
   y = 12x - 3
   \]

### Final Answer:
The equation of the tangent line is:
\[
y = 12x - 3
\]

Would you like a deeper explanation of any steps? 

Here are 5 related questions for further practice:
1. How do you find the tangent line to a function given only its graph?
2. How would you find the normal line to a curve at a given point?
3. Can you find the second derivative of the given function?
4. What happens to the slope of the tangent line as \( x \) approaches infinity for the function \( y = 3x^4 - 42 \)?
5. How do you interpret the geometric meaning of a derivative at a given point?

**Tip:** When simplifying functions, always check for common terms that can cancel out, as this often makes differentiation easier!

Find the equation of the line that is tangent to the graph of the function y = 3x^4 - 6(7/x) at the point (1, 9).

Solve a calculus problem by finding the equation of the tangent line to the function y = 3x^4 - 6(7/x) at the point (1,9). This step-by-step solution covers differentiation and tangent line equations, making it suitable for students in Grades 11-12.

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