Problem 4 - Calculus and its applications

Question

Differentiate. $$ f(x)=15^{x} $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 15^x \ln(15)\).

Step 1: Identify the differentiation rule applicable

To differentiate the function \(f(x) = 15^x\), we use the formula for the derivative of an exponential function. The general rule is that the derivative of \(a^x\) with respect to \(x\) is \(a^x \ln(a)\). This is because \(a^x\) can be rewritten using the base \(e\) as \(e^{x \ln(a)}\).

Step 2: Differentiate the function

Apply the differentiation rule for exponential functions. Here \(a = 15\), so the derivative of \(15^x\) with respect to \(x\) is \(15^x \ln(15)\). Therefore, the derivative \(f'(x) = 15^x \ln(15)\).

Key Concepts

Exponential Functions

Exponential functions are a key topic in calculus and mathematical analysis. They are characterized by a constant base raised to a variable exponent. In functions like \(f(x) = 15^x\), the base is a constant, 15, and the exponent is the variable \(x\). Exponential functions are widely used to model situations where growth or decay is proportional to the current amount, such as in population dynamics, radioactive decay, or compound interest. When differentiating an exponential function, it’s essential to understand how the transformation occurs from the original form to the base \(e\). An exponential function \(a^x\) can be expressed using the natural base \(e\) as \(e^{x \ln(a)}\). This transformation lays the groundwork for applying the rule of differentiation to such functions.

Differentiation

Differentiation is a key process in calculus that finds the rate at which a function is changing at any given point. It is the cornerstone of calculus, used to compute velocities, slopes, and rates of change in various applications. When differentiating, the goal is to derive a new function that represents this rate of change.For exponential functions like \(f(x) = 15^x\), differentiation involves applying specific rules to find \(f'(x)\), the derivative. This process helps in determining how the function behaves across different values of \(x\). The derivative gives insights into the function's increasing or decreasing nature, and how sharply these rates differ at various points.

Derivative Rules

Derivative rules are formulas that provide the derivatives of specific types of functions. These rules simplify the process of differentiation, avoiding the need to revert to first principles each time. A key rule for exponential functions is that if you have \(a^x\), its derivative is \(a^x \ln(a)\). This rule stems from writing the original function in terms of base \(e\) and applying what is known as the chain rule.The natural logarithm \(\ln(a)\) in this rule arises because of the conversion to the base \(e\) and its unique properties. Base \(e\), being approximately 2.718, is inherently linked to continuous growth and decay processes. In the solution for \(f(x) = 15^x\), applying this derivative rule means simply multiplying the function by \(\ln(15)\) to get \(f'(x) = 15^x \ln(15)\). These derivative rules allow us to easily understand and predict how exponential functions change.