Question

Compute the derivative of the given function. $$m(w)=\frac{3^{w}}{2^{w}}$$

Short answer

The derivative is \( m'(w) = \left(\frac{3}{2}\right)^w \ln\left(\frac{3}{2}\right) \).

Step-by-step solution

Express the Function Using Exponents

The given function is written as a quotient of two exponential functions: \[ m(w) = \frac{3^w}{2^w} \]This can be expressed as a single exponential function:\[ m(w) = \left(\frac{3}{2}\right)^w \] This step helps us simplify the function by recognizing it as an exponential with base \(\frac{3}{2}\).

Use the Exponential Function Derivative Formula

The derivative of an exponential function \(a^w\) with respect to \(w\) is given by:\[ \frac{d}{dw}a^w = a^w \ln(a) \]In our case, the base \(a\) is \(\frac{3}{2}\).

Apply the Formula

Using the formula from Step 2, we differentiate \(m(w)\):\[ m'(w) = \left(\frac{3}{2}\right)^w \ln\left(\frac{3}{2}\right) \]We have taken the derivative of \( \left(\frac{3}{2}\right)^w \) using the rule for exponentials.

Key concepts

Exponential Functions

Exponential functions are a key aspect of mathematics, especially in calculus and algebra. They describe processes that change rapidly and are often used to model growth or decay. An exponential function is typically written in the form \( a^x \), where \( a \) is a constant and \( x \) is the variable. Exponents like these are found in various fields such as:

• Biology (to model populations)
• Finance (to calculate compound interest)
• Physics (to understand radioactive decay)

In the context of our function, \( m(w) = \left(\frac{3}{2}\right)^w \), we can recognize this as an exponential growth function with a base greater than 1, i.e., \( \frac{3}{2} \). This means that for each unit increase in \( w \), the value of the function multiplies by \( \frac{3}{2} \). Understanding the behavior of exponential functions is crucial for solving different mathematical problems efficiently.

Differentiation

Differentiation is a fundamental concept in calculus that concerns the calculation of a derivative. A derivative represents how a function changes as its input changes, often thought of as the slope of the function at any given point.
When we differentiate an exponential function, there is a special rule: if the function is of the form \( a^x \), its derivative is \( a^x \ln(a) \). This is particularly useful because it allows us to understand the rate of change of exponential functions.
In the case of \( m(w) = \left(\frac{3}{2}\right)^w \), applying differentiation gives us \( m'(w) = \left(\frac{3}{2}\right)^w \ln\left(\frac{3}{2}\right) \). This expression tells us how quickly the function \( m(w) \) is increasing at any point \( w \). Differentiation is therefore essential for understanding dynamics in systems where quantities change rapidly or subtly.

Calculus

Calculus is a branch of mathematics that studies continuous change and is divided into two main parts: differentiation and integration. Both components are essential for a deep understanding of mathematical modeling.
Differentiation, as previously discussed, provides insight into the rate of change, while integration deals with the accumulation of quantities. Together, they form the basis for analyzing many real-world problems, from engineering to economics.
In the example given with \( m(w) = \left(\frac{3}{2}\right)^w \), calculus allows us to not only differentiate the function but also potentially integrate it to find areas under the curve or accumulative values.

• Practical applications: Calculus helps us calculate exact values that predict future trends.
• Dynamic Systems: It helps in understanding systems that are inherently changeable over time.

Mastering the tools of calculus opens doors to solving complex and dynamic problems seamlessly.