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We are tasked with determining the equation of the tangent line to the curve \( y = e^x \cos(x) \) at the point \( (0,1) \).

### Step 1: Find the derivative \( \frac{dy}{dx} \)
Since we need the slope of the tangent line, we first compute the derivative of the function \( y = e^x \cos(x) \). We will use the **product rule**, as the function is a product of two terms, \( e^x \) and \( \cos(x) \).

The product rule states that:
\[
\frac{d}{dx}(uv) = u'v + uv'
\]
where \( u = e^x \) and \( v = \cos(x) \).

Now, differentiate:
- \( u' = \frac{d}{dx}(e^x) = e^x \)
- \( v' = \frac{d}{dx}(\cos(x)) = -\sin(x) \)

Using the product rule:
\[
\frac{dy}{dx} = e^x \cos(x) + e^x (-\sin(x)) = e^x (\cos(x) - \sin(x))
\]

### Step 2: Evaluate the derivative at \( x = 0 \)
Now, we need to find the slope of the tangent line at \( x = 0 \). Substituting \( x = 0 \) into the derivative:

\[
\frac{dy}{dx}\Bigg|_{x=0} = e^0 (\cos(0) - \sin(0)) = 1(1 - 0) = 1
\]

Thus, the slope of the tangent line at \( (0, 1) \) is \( 1 \).

### Step 3: Use the point-slope form of the equation of a line
The equation of a tangent line can be written using the **point-slope form**:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency.

Here, \( m = 1 \), \( x_1 = 0 \), and \( y_1 = 1 \). Substituting these values into the formula:
\[
y - 1 = 1(x - 0)
\]
Simplifying:
\[
y = x + 1
\]

### Final Answer:
The equation of the tangent line to the curve \( y = e^x \cos(x) \) at the point \( (0,1) \) is:
\[
y = x + 1
\]

Would you like a further breakdown of any step or additional details? 

Here are some related questions you might want to explore:
1. How do you apply the product rule in differentiation?
2. What are the key features of the point-slope form of a line equation?
3. How would the equation of the tangent line change if the point of tangency was different?
4. What is the significance of the derivative at a point in determining tangency?
5. Can the second derivative tell us anything about the curvature of this function?

**Tip:** Always simplify trigonometric values like \( \cos(0) \) and \( \sin(0) \) before evaluating derivatives at specific points to make calculations easier.

Determine the equation of the tangent line to the curve y = e^x cos(x) at point (0,1).

Find the equation of the tangent line to the curve y = e^x cos(x) at the point (0,1). This solution involves key calculus concepts such as derivatives, the product rule, and the equation of a tangent line. Suitable for Grades 11-12.

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