Problem 20 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=B+A e^{t}\)

Step-by-Step Solution

Answer

The derivative of the function is \(\frac{dy}{dt} = A e^t\).

Step 1: Understanding Differentiation

When differentiating a function, we aim to find the derivative, which represents the rate of change of the function. For the function given, we will differentiate with respect to the variable \(t\).

Step 2: Function Components Identification

The function is \(y = B + A e^t\). Here, \(B\) is a constant and \(A e^t\) is an exponential function. The derivative of a constant is zero, and the derivative of \(e^t\) is \(e^t\).

Step 3: Applying Derivative Rules

According to the differentiation rules, the derivative of a constant \(B\) is \(0\). The derivative of \(A e^t\) is \(A \cdot e^t\), since \(A\) is a constant multiplier that remains unaffected by differentiation.

Step 4: Calculating the Derivative

Differentiating the entire function: 1. The derivative of \(B\) is \(0\). 2. The derivative of \(A e^t\) is \(A e^t\). Therefore, the derivative of \(y = B + A e^t\) is \(\frac{dy}{dt} = A e^t\).

Key Concepts

Exponential Function

In mathematics, an exponential function is one where a constant base is raised to a variable exponent. A common example is the function \( e^t \), where \( e \) is Euler's number, approximately 2.71828. This unique constant is significant because it naturally arises in various areas, including calculus, finance, and natural systems. Exponential functions are known for their rapid growth or decay, depending on the sign of the exponent.
An important feature of the exponential function \( e^x \) is that it is its own derivative. This means that if you differentiate \( e^t \), it remains \( e^t \). This property makes it very convenient to work with, especially in problems involving growth models or compounding processes.
In our example, \( y = B + A e^t \), the term \( A e^t \) is where the exponential function comes into play. Understanding this is key to differentiating the function correctly.

Derivative Rules

When differentiating functions, it is essential to use specific rules to find derivatives efficiently. Here are some key rules that we often use:

• Constant Rule: The derivative of a constant is always zero, because constants do not change.
• Power Rule: The derivative of \( x^n \) (where \( n \) is any real number) is \( nx^{n-1} \).
• Exponential Rule: For functions of the form \( e^x \), the derivative is \( e^x \). If there is a constant multiplier, as in \( A e^t \), the derivative is simply \( A \cdot e^t \).

In the original exercise, these rules help simplify the differentiation of \( y = B + A e^t \). Applying the constant rule results in a zero derivative for the constant \( B \). The exponential rule tells us that \( A e^t \) differentiates to \( A e^t \), simplifying the task considerably.

Constant and Variable Differentiation

Differentiation involves determining how a function changes with respect to its variables. It is crucial to distinguish between constants and variables to apply the correct differentiation techniques.

• Constants: Values that remain the same (like \( B \) and \( A \) in our example). Their derivatives are zero as they do not vary with the changes in variables.
• Variables: Values that can change (such as \( t \) in the example \( y = B + A e^t \)). Their changes are captured by taking derivatives.

In our example, the function is composed of both constants and a variable exponent, requiring careful application of the differentiation rules. Understanding the difference between constant and variable parts of a function is fundamental to successfully finding a derivative. By viewing \( t \) as the variable, we use its properties to determine the rate of change of the function \( y \).