Problem 22 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(5^{x}\)

Step-by-Step Solution

Answer

The derivative of \( 5^x \) is \( 5^x \ln(5) \).

Step 1: Understand the Exponential Function Derivative

For any exponential function of the form \( a^x \), the derivative with respect to \( x \) is given by \( a^x \ln(a) \). Here, \( a \) is the base of the exponential function.

Step 2: Identify Parameters of the Function

In the given function \( 5^x \), the base \( a \) is 5. We will use this in the derivative formula \( a^x \ln(a) \).

Step 3: Apply the Derivative Formula

Using the formula from Step 1 and the identified parameters, substitute \( a = 5 \) into the derivative formula: \[ \frac{d}{dx}[5^x] = 5^x \ln(5) \] This provides the derivative of the function.

Step 4: Write the Final Result

The derivative of the expression \( 5^x \) with respect to \( x \) is \( 5^x \ln(5) \). This is the simplified version of the expression after differentiation.

Key Concepts

Exponential Function

An exponential function is a mathematical expression of the form \( a^x \), where \( a \) is a constant base and \( x \) is the variable exponent. Exponential functions are unique because the variable is the exponent, rather than the base. This is a key distinction that gives exponential functions their distinctive growth characteristics. Main Points:

• Base \( a \): The base \( a \) must be a positive real number. Typically, \( a > 0 \) and \( a eq 1 \).
• Rapid Growth or Decay: Depending on the base, exponential functions can grow or decay rapidly. For example, if \( a > 1 \), the function exhibits exponential growth. Conversely, if \( 0 < a < 1 \), it shows exponential decay.
• Common Bases: Often, the base is \( e \) (approximately 2.718) or 10. However, any positive number can serve as the base, such as in the problem where base 5 is used.

Understanding these points helps you recognize the nature of the function you're differentiating, which is crucial, especially when working with more complex expressions.

Exponential Derivatives

Differentiating exponential functions involves special derivative rules. For an exponential function with a base \( a \), the derivative essentially mimics the structure of the function itself.Formula for Differentiation:

• The general formula for the derivative of \( a^x \) is \( a^x \ln(a) \).

Here's why this works:

• Chain Rule: The differentiation of \( e^x \) is \( e^x \) itself. Thus, for \( a^x \), rewrite it using \( e \): \( a^x = e^{x \ln a} \).
• Application of Chain Rule: Differentiating \( e^{x \ln a} \) by applying the chain rule, we get \( \ln(a) \cdot e^{x \ln a} = a^x \ln(a) \).

This reveals why the derivative \( a^x \ln(a) \) preserves the exponential form, scaled by the natural logarithm of the base \( a \). Recognizing these derivative characteristics simplifies processes for more complicated functions.

Derivative Formulas

Derivative formulas are the basis of calculus problem-solving, used to find rates at which things change. The general understanding of derivative formulas, especially for exponential functions, is vital for performing accurate calculations. Key Derivative Formulas:

• Exponential Functions: As previously mentioned, the derivative of \( a^x \) is \( a^x \ln(a) \).
• Natural Exponential Function: The derivative of \( e^x \) is simple since it is \( e^x \) itself, facilitating calculations in a variety of contexts.
• Power Rule: For any power function \( x^n \), the derivative is \( nx^{n-1} \).
• Sum Rule: If you differentiate a sum of functions, you can do so individually for each function and then sum those derivatives.

These formulas are vital for understanding how changes in one variable can affect another. By mastering these derivative concepts, you can tackle not only textbook exercises but also complex real-world mathematical problems.