Problem 10 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=5 \cdot 2^{x}-5 x+4\)

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 5 \cdot 2^x \ln(2) - 5 \).

Step 1: Identify the function components

The function given is a combination of exponential and polynomial terms: \( y = 5 \cdot 2^x - 5x + 4 \). Here, \( 5 \cdot 2^x \) is an exponential term and \( -5x + 4 \) is a linear polynomial term.

Step 2: Differentiate the exponential term

The derivative of \( 5 \cdot 2^x \) is calculated using the rule for differentiating exponential functions: \( \frac{d}{dx}[a \cdot b^x] = a \cdot b^x \ln(b) \). Therefore, the derivative of \( 5 \cdot 2^x \) is \( 5 \cdot 2^x \ln(2) \).

Step 3: Differentiate the polynomial term

The polynomial \( -5x + 4 \) comprises a linear term \( -5x \) and a constant \( 4 \). The derivative of \( -5x \) is \( -5 \), and the derivative of a constant \( 4 \) is \( 0 \).

Step 4: Combine derivatives

Combining the derivatives from the previous steps gives the derivative of the entire function \( y \). Thus, \( \frac{dy}{dx} = 5 \cdot 2^x \ln(2) - 5 \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, where we focus on finding the rate at which a function changes. It's like figuring out how fast something is moving at any given moment. In more technical terms, it's about finding the derivative of a function.

To differentiate a function like the one in our exercise, we look at each term separately. When you have a combination of different kinds of functions, such as exponential and polynomial, you apply differentiation rules specific to each. The process often involves applying these rules:

• Power Rule: For any function of the form \(cf(x)^n\), its derivative is \(cnf(x)^{n-1}\).
• Constant Rule: The derivative of a constant is zero.
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

This allows us to break down the differentiation task into smaller, more manageable parts. Differentiating each term of our function results in a combination of separate derivatives that give us the rate of change of the entire function.

Understanding differentiation requires practice, but once you master the basic rules, it becomes easy to apply them to complex functions.

Exponential functions

Exponential functions might sound complex, but they're really about growth and decay. In mathematics, these functions are characterized by having a constant raised to the power of a variable. The classic form looks like \(a \cdot b^x\), where \(a\) is a constant, and \(b\) is the base of the exponential.

One of their key features is that they grow or shrink very rapidly depending on whether the base \(b\) is greater than or less than 1. For the differentiation of exponential functions, there is a specific rule: The derivative of \(a \cdot b^x\) is \(a \cdot b^x \ln(b)\). This works because of the natural logarithm \(\ln\), which is crucial in helping us scale the rate of growth correctly.

In our exercise, we applied this rule to differentiate \(5 \cdot 2^x\). Remember, exponential functions appear in many natural phenomena, like population growth and radioactive decay, due to their unique rates of change. Recognizing and differentiating them is an essential skill in calculus.

Polynomial functions

Polynomial functions are simpler, more straightforward than exponential functions. They consist of terms whe each variable is raised to a non-negative integer power. For example, a function like \(-5x + 4\) is a linear polynomial because it has the variable \(x\) to the power of one and a constant term.

Differentiating polynomial functions involves applying the power rule. For a term \(cx^n\), the derivative is \(cnx^{n-1}\). This rule simplifies the process because every term can be handled individually.

In our example, differentiating the polynomial part \(-5x + 4\), we see that the derivative of \(-5x\) is \(-5\) since the derivative of \(x\) is 1. The derivative of a constant \(4\) is simply zero, as constants don't change. Combining these results gives the rate of change for the polynomial function.

Polynomials are foundational in mathematics and appear in various scenarios, making their simplicity critical to understanding more complex functions.