Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=2 e^{x}$$

Short answer

The derivative of the function \(y = 2e^{x}\) with respect to \(x\) is \(2e^{x}\)

Step-by-step solution

Identify the function

The function given is \(y = 2e^{x}\). This function is comprised of the independent variable \(x\), the dependent variable \(y\), and the constant \(2\). The exponent \(x\) in the term \(e^{x}\) is a variable as well.

Apply the derivative rule

The derivative rule states that the derivative of \(e^{x}\) with respect to \(x\) is \(e^{x}\). Now you apply this rule to the function \(y = 2e^{x}\). Since the constant multiplier (2) does not affect the derivative, the derivative of \(2e^{x}\) with respect to \(x\) is \(2e^{x}\).

State the final answer

The derivative of the given function \(y = 2e^{x}\) with respect to \(x\) is \(2e^{x}\).

Key concepts

Differential Calculus

Differential calculus is the branch of calculus concerned with the study of how functions change when their inputs change. It's built around the concept of the derivative, which measures the rate at which a function's value changes at any point. For example, in the exercise where we find the derivative of the function y = 2e^{x}, we are essentially calculating how quickly the value of y changes with respect to a change in x.

The process of finding a derivative is known as differentiation. In this straightforward example, the derivative of e^{x} is itself e^{x}, and because differentiation is linear, we can multiply this by the constant 2 to get the derivative of the entire function y, which is 2e^{x}. This idea of differentiation is crucial to understanding the behavior of functions, especially when we want to determine instantaneous rates of change or the slope of the graph at any point.

Exponential Growth and Decay

Exponential growth and decay are processes that increase or decrease at rates proportional to their current size. This is represented mathematically by functions of the form y = Ae^{kx}, where A represents the initial amount, k is a constant that determines the rate of growth (if positive) or decay (if negative), and x is the variable.

In the function y = 2e^{x} from our exercise, we see exponential growth since the base of the exponent is the natural number e, which is approximately 2.71828, and the exponent is the variable x. As x increases, e^{x} grows rapidly, and since it's multiplied by 2, the rate of growth for y is twice that of e^{x}. This behavior is typical of exponential functions and is important in modeling situations where growth accelerates over time, such as populations, investments, and certain physical processes.

Chain Rule

The chain rule is a powerful tool in differential calculus for finding the derivative of composite functions. When a function y is composed of another function of x, the chain rule allows us to differentiate y with respect to x by multiplying the derivative of y with respect to the inner function by the derivative of the inner function with respect to x.

However, in our exercise y = 2e^{x}, the chain rule isn't required since it's a simple exponential function without an inner function more complex than x. If we had a more complicated exponent, such as a polynomial or another function of x, the chain rule would be absolutely necessary to find the derivative. Understanding how to apply the chain rule is essential for tackling a wide array of problems in calculus, especially when dealing with nested functions where one function is dependent on another.