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Magazine Chapter 4: Problem 52
Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(r)=4^{r^{1 / 4}} $$
Short Answer
Expert verified
The derivative is \( g'(r) = \frac{1}{4} \ln(4) \cdot 4^{r^{1/4}} \cdot r^{-3/4} \).
Step by step solution
01
Identify the Function
We have the function \( g(r) = 4^{r^{1/4}} \). Our task is to differentiate this function with respect to \( r \).
02
Apply the Chain Rule
To differentiate \( g(r) = 4^{r^{1/4}} \), we can use the chain rule. Let \( u = r^{1/4} \), so \( 4^{r^{1/4}} = 4^u \). We differentiate \( g(r) \) as the composition of the functions \( 4^u \) and \( u = r^{1/4} \).
03
Differentiate the Exponential Function
The derivative of \( 4^u \) with respect to \( u \) is \( 4^u \, \ln(4) \). Therefore, if \( g(u) = 4^u \), then \( g'(u) = 4^u \ln(4) \).
04
Differentiate the Inner Function
Next, differentiate \( u = r^{1/4} \) with respect to \( r \). The derivative is \( \frac{1}{4} r^{-3/4} \).
05
Combine Using Chain Rule
By the chain rule, the derivative of \( g(r) = 4^{r^{1/4}} \) with respect to \( r \) is the product of \( g'(u) = 4^u \ln(4) \) and \( \frac{du}{dr} = \frac{1}{4} r^{-3/4} \). Substitute \( u = r^{1/4} \) back into \( g'(u) \). The derivative is \( g'(r) = 4^{r^{1/4}} \ln(4) \cdot \frac{1}{4} r^{-3/4} \).
06
Simplify the Expression
The expression for the derivative is \( g'(r) = \frac{1}{4} \ln(4) \cdot 4^{r^{1/4}} r^{-3/4} \). This can be written as \( g'(r) = \frac{\ln(4)}{4} \cdot 4^{r^{1/4}} \cdot r^{-3/4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. Composite functions are functions that are formed by combining two or more functions. For example, in the function \( g(r) = 4^{r^{1/4}} \), there is an outer function \( 4^u \) and an inner function \( r^{1/4} \).
By using the chain rule, we can take the derivative of this composite function by first differentiating the outer function with respect to the inner function, and then multiplying it by the derivative of the inner function with respect to the original variable.
In this case, the chain rule helps us find \( g'(r) \) by expressing it as a product of derivatives.
By using the chain rule, we can take the derivative of this composite function by first differentiating the outer function with respect to the inner function, and then multiplying it by the derivative of the inner function with respect to the original variable.
- Take the derivative of the outer function with respect to the inner function.
- Multiply by the derivative of the inner function with respect to the original variable.
In this case, the chain rule helps us find \( g'(r) \) by expressing it as a product of derivatives.
Exponential Function
Exponential functions are functions of the form \( a^x \), where \( a \) is a constant and \( x \) is the variable. These functions have unique properties and are widely used in calculus and real-life applications.
The specific function in our exercise, \( 4^{r^{1/4}} \), is an example of an exponential function where the base is 4 and the exponent is \( r^{1/4} \). Exponential functions grow rapidly and have a constant rate of change, which makes their derivatives interesting and vital to understand.
For our function, differentiating \( 4^{r^{1/4}} \) means applying the rule for exponential derivatives within the context of the chain rule.
The specific function in our exercise, \( 4^{r^{1/4}} \), is an example of an exponential function where the base is 4 and the exponent is \( r^{1/4} \). Exponential functions grow rapidly and have a constant rate of change, which makes their derivatives interesting and vital to understand.
- The derivative of \( a^x \) with respect to \( x \) is \( a^x \ln(a) \).
- They appear in models of growth and decay, compound interest, and more.
For our function, differentiating \( 4^{r^{1/4}} \) means applying the rule for exponential derivatives within the context of the chain rule.
Derivative
The derivative of a function is a measure of how the function's output changes with respect to a change in the input. It is one of the core concepts in calculus, symbolizing the rate of change or the slope of the function at any given point.
In our example, finding the derivative \( g'(r) \) of \( g(r) = 4^{r^{1/4}} \) helps us understand how fast or slow this function changes as \( r \) changes. The resulting derivative provides insights into the function's behavior, including its increasing or decreasing nature.
Understanding derivatives is essential for tackling problems in physics, engineering, and economics.
In our example, finding the derivative \( g'(r) \) of \( g(r) = 4^{r^{1/4}} \) helps us understand how fast or slow this function changes as \( r \) changes. The resulting derivative provides insights into the function's behavior, including its increasing or decreasing nature.
- Derivatives are used in a variety of fields to analyze dynamic systems.
- The derivative of a composite function involves multiple steps like using the chain rule.
Understanding derivatives is essential for tackling problems in physics, engineering, and economics.
Calculus
Calculus is the branch of mathematics that studies how quantities change. It is divided into two main branches: differential calculus, which involves derivatives, and integral calculus, which involves integrals.
Our problem focuses on differential calculus, which is concerned with finding the rate at which things change. This is crucial for understanding real-world phenomena that involve variable speeds, growth rates, and more.
The exercise exemplifies how calculus, and specifically the derivative concept, can be applied to understand and predict the behavior of mathematical functions.
Our problem focuses on differential calculus, which is concerned with finding the rate at which things change. This is crucial for understanding real-world phenomena that involve variable speeds, growth rates, and more.
- Calculus allows us to solve complex problems involving motion, growth, and patterns.
- It provides tools for modeling and solving equations that describe changing systems.
The exercise exemplifies how calculus, and specifically the derivative concept, can be applied to understand and predict the behavior of mathematical functions.
