Consider the curve `y =e^(2x)`
What is the slope of the tangent to the curve at (0,1) ?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To find the slope of the tangent to the curve \( y = e^{2x} \) at the point \( (0, 1) \), we need to follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( y = e^{2x} \) with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx}(e^{2x}) \]

Using the chain rule, we differentiate:

\[ \frac{dy}{dx} = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x} \]

### Step 2: Evaluate the derivative at \( x = 0 \)
Next, we need to find the slope of the tangent line at the point \( (0, 1) \). We do this by substituting \( x = 0 \) into the derivative we found.

\[ \frac{dy}{dx} \bigg|_{x=0} = 2e^{2 \cdot 0} = 2e^{0} \]

Since \( e^{0} = 1 \):

\[ \frac{dy}{dx} \bigg|_{x=0} = 2 \cdot 1 = 2 \]

### Step 3: Conclusion
Thus, the slope of the tangent to the curve \( y = e^{2x} \) at the point \( (0, 1) \) is \( 2 \).

### Final Answer
The slope of the tangent to the curve at the point \( (0, 1) \) is \( 2 \).

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