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Chapter 3: Problem 4

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

Short Answer

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

Step by step solution

01

- Identify the Function Type

Recognize that the given function, \(f(x) = 15^x\), is an exponential function with a base other than the natural number \(e\).
02

- Recall the Derivative of an Exponential Function

The derivative of an exponential function with base \(a\) is given by the formula: \(\frac{d}{dx}(a^x) = a^x \ln(a)\). Here \(a = 15\).
03

- Apply the Derivative Formula

Using the formula, differentiate the function \(f(x) = 15^x\). This gives: \(f'(x) = 15^x \ln(15)\).
04

- Write the Final Answer

Combine the results to get the final derivative: \(f'(x) = 15^x \cdot \ln(15)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential functions
Exponential functions are a fundamental concept in calculus and algebra. They are of the form \(a^x\), where the base \(a\) is a constant, and the exponent \(x\) is a variable. These functions grow or decay at a rate proportional to their current value.
When differentiating exponential functions, it is crucial to know the base. If the base is the natural number \(e\) (approximately 2.71828), the function simplifies to \(f(x) = e^x\). For other bases, such as in our example function \(f(x) = 15^x\), the differentiation involves the natural logarithm of the base.
differentiation formulas
Differentiation formulas provide us with rules to find the derivative of various functions. For exponential functions, there are specific rules to follow. The general formula for differentiating an exponential function with base \(a\) is:

\[ \frac{d}{dx}(a^x) = a^x \ln(a) \]
This formula states that to find the derivative, you multiply the original function by the natural logarithm of the base \(a\). In our example function, \(f(x) = 15^x\), we used this formula to find its derivative.
First, recognize the function as an exponential function with base \(a = 15\). Then apply the differentiation formula to get: \[ f'(x) = 15^x \ln(15) \]
natural logarithm
The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base \(e\). It has key properties that make it extremely useful in calculus:
  • It is the inverse of the exponential function with base \(e\), such that \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\).
  • It helps simplify the differentiation of exponential functions. For example, when finding \(\frac{d}{dx}(a^x)\), we need the natural logarithm of the base \(a\).

In our differentiation problem, the natural logarithm of 15, \(\ln(15)\), is part of the final derivative formula. This highlights the connection between exponential functions and natural logarithms in calculus.
Using these properties can make complex calculus problems more manageable and intuitive.

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