Problem 34 - Calculus Single Variable

Question

Calculate the derivative of the given xpression with respect to \(x\). $$ 5^{2-x} $$

Step-by-Step Solution

Answer

The derivative is \(-5^{2-x} \ln 5\).

Step 1: Identify the Derivative Form

To find the derivative of an expression of form \(a^u\) where \(u\) is a function of \(x\), we use the formula \(\frac{d}{dx}a^u = a^u \ln a \cdot \frac{du}{dx}\). In this case, \(a = 5\) and \(u = 2 - x\).

Step 2: Calculate the Derivative of the Exponent

Calculate \(\frac{du}{dx}\) for \(u = 2 - x\). Since \(u = 2 - x\), its derivative is \(\frac{d}{dx}(2 - x) = -1\).

Step 3: Apply the Derivative Formula

Substitute into the formula: \(\frac{d}{dx}5^{2-x} = 5^{2-x} \ln 5 \cdot (-1)\).

Step 4: Simplify the Expression

Simplify the expression to get \(-5^{2-x} \ln 5\).

Key Concepts

Exponential Functions

Exponential functions involve expressions where the variable is in the exponent. They have the form \(a^x\), where \(a\) is a constant and \(x\) is the variable. These functions grow or decay at rates proportional to their current value, meaning that the change is exponential rather than linear. Exponential functions are prevalent in many fields such as biology, finance, and physics because they naturally describe growth processes, decay, and various other processes.In the context of the exercise, the function \(5^{2-x}\) illustrates an exponential function where the base \(5\) is raised to the power of \((2-x)\). The '-x' in the exponent signifies that as \(x\) increases, the power decreases, and consequently, the entire expression represents exponential decay. Understanding how such functions behave is crucial for tackling derivatives because these can often be involved in growth models, decay processes, and financial projections.

Differentiation Rules

Differentiation rules provide the framework and techniques needed to find the derivative of a function. Derivatives measure how a function changes as its input changes. For exponential functions, knowing the right rules is essential since these functions involve variables located in the exponent position, requiring specific strategies for differentiation.Several important rules are usually applied, such as:

• Power Rule: Used for functions of the form \(x^n\), it states that the derivative is \(nx^{n-1}\).
• Chain Rule: Useful when dealing with composite functions. If a function \(y\) can be expressed as \(y = g(f(x))\), the derivative is found using \(g'(f(x)) \cdot f'(x)\).
• Exponential Rule: Applied specifically to exponential functions, taking into account the unique nature of exponential growth or decay.

In the provided problem, the differentiation of \(5^{2-x}\) requires both knowledge of the exponential rule and the chain rule to account for the expression in the exponent. The step-by-step solution employs these rules to simplify and correctly derive the answer.

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate complex functions, especially those involving products, quotients, or variable exponents. By employing the properties of logarithms, one can simplify differentiation tasks involving exponential functions.When a function has an exponent that is another function of \(x\), taking the natural logarithm of both sides facilitates easier differentiation. This method leverages properties such as:

• The natural logarithm of a power simplifies: \(\ln(a^b) = b \cdot \ln(a)\).
• The derivative of \(\ln(x)\) is \(\frac{1}{x}\).

However, in the original exercise, direct exponential differentiation is used, which is more straightforward for simple exponential forms. When the complexities increase, logarithmic differentiation becomes an invaluable tool. Understanding when to apply each method can make the differentiation process much more manageable and efficient, reducing potential errors.