91Ó°ÊÓ

The derivative is a fundamental concept in calculus. It represents the rate at which a function changes. For the function \( f(x) = e^{2x} \), to find its derivative, we use the chain rule. The chain rule states that the derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \). For our function, \( u = 2x \), so \( \frac{du}{dx} = 2 \). This gives us the derivative of \( f(x) \) as \( f'(x) = 2e^{2x} \).

Here are the essential steps in deriving this function:

• Identify the inner function: In \( e^{2x} \), the inner function is \( 2x \).
• Differentiate the inner function: \( \frac{d}{dx} (2x) = 2 \).
• Multiply by the derivative of the inner function: \( e^{2x} \cdot 2 \).

The derivative signifies the slope of the tangent line to the curve at any given point along the curve. For our function, knowing that the derivative \( 2e^{2x} \) is always positive means that the function is increasing everywhere.

Local extrema are points where a function reaches a local maximum or minimum. It's where the function "levels out" momentarily before changing direction. However, to find these points, the derivative needs to be zero or undefined at some point.

In the case of \( f(x) = e^{2x} \), the derivative \( f'(x) = 2e^{2x} \) is always positive. It never equals zero and is never undefined. Therefore, no local extrema exist for this function. This means that there are no peaks or troughs on the graph.

To summarize:

• Local extrema occur where \( f'(x) = 0 \) or is undefined.
• For \( e^{2x} \), \( f'(x) = 2e^{2x} \) is always positive.
• Thus, no local maxima or minima are present in the function.

Understanding where a function is increasing or decreasing helps visualize its behavior. A function is increasing if its derivative is positive, while it is decreasing if the derivative is negative.

For \( f(x) = e^{2x} \), the derivative \( f'(x) = 2e^{2x} \) is positive for all real values of \( x \). Thus, the function is increasing on the entire domain of all real numbers. This means the curve of \( f(x) \) continuously rises as \( x \) increases.

Key points to note:

• An increasing function has \( f'(x) > 0 \).
• A decreasing function would have \( f'(x) < 0 \). However, \( f'(x) = 2e^{2x} \) never becomes negative.
• Therefore, \( f(x) = e^{2x} \) increases for all \( x \).

This concept helps predict the behavior of functions and their graphs, ensuring a solid understanding of calculus principles.