Problem 7 - Calculus

Question

Find the derivatives of the functions $$ (2 / 3)^{x} $$

Step-by-Step Solution

Answer

The derivative is \(\left(\frac{2}{3}\right)^x \ln\left(\frac{2}{3}\right)\).

Step 1: Understand the Function

The function we need to find the derivative of is \(f(x) = \left(\frac{2}{3}\right)^x\). This is an exponential function.

Step 2: Use the Derivative of an Exponential Function Formula

For a general exponential function \(a^x\), the derivative is \(a^x \ln(a)\), where \(a\) is a constant and \(\ln\) is the natural logarithm.

Step 3: Apply the Formula

Here, \(a = \frac{2}{3}\). Substitute \(a = \frac{2}{3}\) into the derivative formula: \(\frac{d}{dx}\left(\left(\frac{2}{3}\right)^x\right) = \left(\frac{2}{3}\right)^x \ln\left(\frac{2}{3}\right)\).

Step 4: Simplify the Expression

The derivative of \(f(x) = \left(\frac{2}{3}\right)^x\) is therefore: \(f'(x) = \left(\frac{2}{3}\right)^x \ln\left(\frac{2}{3}\right)\).

Key Concepts

Natural Logarithm

When dealing with exponential functions, the natural logarithm, often represented as \(\ln\), plays a crucial role, especially in calculus. Unlike common logarithms, which are base 10, the natural logarithm uses Euler's number \(e\) as its base. Euler's number \(e\) is an important mathematical constant approximately equal to 2.71828. The natural logarithm is denoted by \(\ln(x)\) and represents the power to which \(e\) must be raised to obtain \(x\).Understanding natural logarithms is essential when differentiating exponential functions such as \(a^x\). The derivative involves multiplying by \(\ln(a)\), which essentially scales the rate of change of the function by this logarithmic factor. This is why knowing how to use the natural logarithm is critical in calculus, as it simplifies and streamlines many differentiation problems, especially involving exponential growth or decay functions.

Exponential Function

Exponential functions can appear daunting, but they are fundamentally straightforward. These functions take the form \(a^x\), where \(a\) is a positive constant, and \(x\) is the variable exponent. A key characteristic of exponential functions is their rapid increase or decrease, depending on the value of \(a\). When \(a > 1\), the function grows exponentially, while if \(0 < a < 1\), it decays.The function in our exercise, \(f(x) = \left(\frac{2}{3}\right)^x\), is an example of exponential decay because \(\frac{2}{3}\) is less than 1. Such functions are pivotal in modeling real-world processes like radioactive decay and population decline. They represent constant multiplicative rates of change, differing from linear functions, which change additively.

Derivative Formula

Derivatives measure how a function changes as its input changes; they are fundamental in calculus. For exponential functions like \(a^x\), the derivative formula is \(a^x \ln(a)\). This formula illustrates the rate of change of the function concerning \(x\), incorporating both the original function and the natural logarithm of the base \(a\). Let's apply this to our function \(f(x) = \left(\frac{2}{3}\right)^x\). By substituting \(a = \frac{2}{3}\) into the derivative formula, we calculate:

• The derivative is \(\left(\frac{2}{3}\right)^x \ln\left(\frac{2}{3}\right)\).
• This derivative shows the function's rate of decay or growth rate, scaled by \(\ln\left(\frac{2}{3}\right)\).
• This also highlights the importance of the derivative formula in analyzing how rapidly an exponential function changes.

Differentiation is a versatile tool for figuring out instantaneous rates of change, crucial for various applications in science and engineering.