Problem 336 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\frac{10^{x}}{\ln 10}$$

Step-by-Step Solution

Answer

The derivative is \(f^{\prime}(x) = 10^x\).

Step 1: Identify the Derivative Rule Needed

To differentiate the function \(f(x) = \frac{10^{x}}{\ln 10}\), we need to use the derivative of the exponential function \(a^{x}\), where \(a\) is a constant. The derivative of \(a^x\) is \(a^x \ln a\). In our case, \(a = 10\).

Step 2: Apply the Derivative Rule

Apply the rule derived in Step 1: \(\frac{d}{dx}[10^x] = 10^x \ln 10\). Now substitute this into the function: \(f^{\prime}(x) = \frac{10^x \ln 10}{\ln 10}\).

Step 3: Simplify the Expression

Notice that the \(\ln 10\) in the numerator and the denominator cancel each other out. Hence, \(f^{\prime}(x) = 10^x\).

Key Concepts

Exponential Function

An exponential function is a type of mathematical function in which a constant base is raised to the power of a variable. This means that as the variable changes, the function's value changes exponentially. A basic example is the function \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent.
This type of function showcases a rate of growth that compounds, making it vital for concepts in natural sciences and financial models. When working with exponential functions where the base is a constant, understanding the behavior of the function is essential when moving forward to differentiation.

Derivative of a Constant Base Exponential

Differentiating exponential functions with a constant base requires a specific rule. For an exponential function of the form \( f(x) = a^x \), the derivative is obtained using the formula \( (a^x)' = a^x \ln a \).
This rule is derived from the properties of logarithms and the natural exponential function, and it is crucial when tackling problems involving growth and decay. In our exercise, the function \( f(x) = \frac{10^x}{\ln 10} \) required us to apply this derivative rule while recognizing that \( a = 10 \). This direct application simplifies the process and allows us to compute derivatives efficiently.

Simplifying Expressions

Simplifying expressions is a crucial step after differentiation, as it provides a clearer, simpler form of the derived function. In our example, after using the differentiation rule, we reached an expression \( f^{\prime}(x) = \frac{10^x \ln 10}{\ln 10} \).
This expression contains a common term in both the numerator and the denominator, specifically \( \ln 10 \). By canceling these terms, the derivative simplifies to \( f^{\prime}(x) = 10^x \).

• The process of simplification not only makes the mathematical expression easier to understand but also highlights the underlying behavior of the function more transparently.
• Being comfortable with simplification techniques is essential for tackling more complex calculus problems.