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The problem asks to find the equation of the tangent line to the curve \( y = x^4 + 6e^x \) at the point \( (0, 6) \).

To find the equation of the tangent line, we follow these steps:

### Step 1: Calculate the derivative \( \frac{dy}{dx} \).
The derivative of the function \( y = x^4 + 6e^x \) gives us the slope of the tangent line. Using standard rules of differentiation:

\[
\frac{dy}{dx} = \frac{d}{dx}(x^4) + \frac{d}{dx}(6e^x)
\]
\[
\frac{dy}{dx} = 4x^3 + 6e^x
\]

### Step 2: Evaluate the derivative at \( x = 0 \).
To find the slope of the tangent line at the point \( (0, 6) \), we substitute \( x = 0 \) into the derivative:

\[
\frac{dy}{dx}\bigg|_{x=0} = 4(0)^3 + 6e^0 = 6
\]

So, the slope of the tangent line at \( x = 0 \) is 6.

### Step 3: Use the point-slope form of a line.
The point-slope form of 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. In this case, the slope \( m = 6 \) and the point is \( (0, 6) \).

Substitute these values into the equation:

\[
y - 6 = 6(x - 0)
\]

Simplifying:

\[
y = 6x + 6
\]

Thus, the equation of the tangent line is:
\[
y = 6x + 6
\]

Would you like more details on any step or have further questions?

Here are 5 related questions to explore:
1. How do you find the second derivative of \( y = x^4 + 6e^x \)?
2. What is the general method to find a tangent line for any given function at a point?
3. How would you find the normal line to this curve at the same point?
4. What is the geometric interpretation of the derivative at a specific point?
5. Can we approximate \( y = x^4 + 6e^x \) around \( x = 0 \) using Taylor series?

**Tip**: Remember that the tangent line provides the best linear approximation of a curve at a specific point.

Find the equation of the tangent line to the curve y = x^4 + 6e^x at the point (0, 6).

Learn how to find the equation of the tangent line to the curve y = x^4 + 6e^x at the point (0, 6) using derivatives and the point-slope formula. Suitable for high school students in Grades 11-12.

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