Applying the Chain Rule to the Problem

Oct 13, 2025 by windailys.com 15 views

Understanding the Problem

In this problem, we are given a differential equation that models the spread of a rumor in a population. The equation is $ p(t)=\frac{1}{1+a e^{-k t}} $, where $ p(t) $ is the proportion of the population that knows the rumor at time $ t $, and $ a $ and $ k $ are constants. Our goal is to find the derivative of $ p(t) $ with respect to $ t $, which represents the rate at which the rumor is spreading.

The Chain Rule

To find the derivative of $ p(t) $, we can use the chain rule, which is a fundamental concept in calculus. The chain rule states that if we have a composite function of the form $ f(g(x)) $, then the derivative of $ f(g(x)) $ with respect to $ x $ is given by $ f'(g(x)) \cdot g'(x) $. In other words, we need to find the derivative of the outer function $ f $ and the derivative of the inner function $ g $, and then multiply them together.

Applying the Chain Rule to the Problem

In this case, we have a composite function of the form $ p(t) = \frac{1}{1+a e^{-k t}} $. To apply the chain rule, we need to identify the outer function and the inner function. The outer function is $ f(x) = \frac{1}{1+x} $, and the inner function is $ g(t) = a e^{-k t} $. We can now find the derivative of the outer function and the inner function, and then multiply them together.

Finding the Derivative of the Outer Function

To find the derivative of the outer function $ f(x) = \frac{1}{1+x} $, we can use the quotient rule, which states that if we have a function of the form $ \frac{u}{v} $, then the derivative of $ \frac{u}{v} $ with respect to $ x $ is given by $ \frac{u'v - uv'}{v^2} $. In this case, we have $ u = 1 $ and $ v = 1+x $, so the derivative of $ f(x) $ is given by:

f'(x) = \frac{(1)'(1+x) - (1)(1+x)'}{(1+x)^2}

Simplifying the expression, we get:

f'(x) = \frac{1 - (1+x)'}{(1+x)^2}

f'(x) = \frac{1 - (1)}{(1+x)^2}

f'(x) = \frac{0}{(1+x)^2}

f'(x) = 0

Finding the Derivative of the Inner Function

To find the derivative of the inner function $ g(t) = a e^{-k t} $, we can use the chain rule again. The derivative of $ e^{-k t} $ is $ -k e^{-k t} $, so the derivative of $ g(t) $ is given by:

g'(t) = a (-k e^{-k t})

g'(t) = -ak e^{-k t}

Finding the Derivative of $ p(t) $

Now that we have found the derivative of the outer function and the inner function, we can multiply them together to find the derivative of $ p(t) $:

p'(t) = f'(g(t)) \cdot g'(t)

p'(t) = 0 \cdot (-ak e^{-k t})

p'(t) = 0

Conclusion

In this problem, we used the chain rule to find the derivative of a composite function that models the spread of a rumor in a population. We identified the outer function and the inner function, and then found the derivative of each function using the quotient rule and the chain rule. Finally, we multiplied the derivatives together to find the derivative of the composite function.

Example Use Case

The chain rule is a powerful tool in calculus that can be used to find the derivative of a wide range of functions. In this example, we used the chain rule to find the derivative of a function that models the spread of a rumor in a population. This type of function is commonly used in epidemiology to model the spread of diseases.

Code Implementation

Here is an example of how to implement the chain rule in Python:

import sympy as sp

# Define the variables
t = sp.symbols('t')
a = sp.symbols('a')
k = sp.symbols('k')

# Define the function
p = 1 / (1 + a * sp.exp(-k * t))

# Find the derivative of the function using the chain rule
p_prime = sp.diff(p, t)

# Print the result
print(p_prime)

This code uses the SymPy library to define the function and find its derivative using the chain rule. The result is printed to the console.

Conclusion

Q: What is the chain rule in calculus?

A: The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. A composite function is a function that is composed of two or more functions, where the output of one function is used as the input of another function.

Q: How do I apply the chain rule to a composite function?

A: To apply the chain rule, you need to identify the outer function and the inner function. The outer function is the function that is being composed, and the inner function is the function that is being used as the input of the outer function. Once you have identified the outer and inner functions, you can find the derivative of each function using the chain rule.

Q: What is the quotient rule, and how is it related to the chain rule?

A: The quotient rule is a rule in calculus that allows us to find the derivative of a quotient of two functions. The quotient rule is related to the chain rule in that it is used to find the derivative of the outer function in the chain rule.

Q: Can you give an example of how to use the chain rule to find the derivative of a composite function?

A: Yes, let's consider the function $ p(t) = \frac{1}{1+a e^{-k t}} $. To find the derivative of this function using the chain rule, we need to identify the outer function and the inner function. The outer function is $ f(x) = \frac{1}{1+x} $, and the inner function is $ g(t) = a e^{-k t} $. We can then find the derivative of each function using the quotient rule and the chain rule, and multiply them together to find the derivative of the composite function.

Q: What are some common mistakes to avoid when using the chain rule?

A: Some common mistakes to avoid when using the chain rule include:

Failing to identify the outer and inner functions correctly
Failing to find the derivative of each function using the correct rule (e.g. quotient rule or chain rule)
Failing to multiply the derivatives together correctly
Failing to simplify the resulting expression

Q: Can you give an example of how to use the chain rule in a real-world application?

A: Yes, let's consider the problem of modeling the spread of a rumor in a population. We can use the chain rule to find the derivative of a composite function that models the spread of the rumor. For example, let's consider the function $ p(t) = \frac{1}{1+a e^{-k t}} $, where $ p(t) $ is the proportion of the population that knows the rumor at time $ t $, and $ a $ and $ k $ are constants. We can use the chain rule to find the derivative of this function, and then use the resulting expression to model the spread of the rumor.

Q: How do I know if I am using the chain rule correctly?

A: To know if you are using the chain rule correctly, you can try the following:

Check that you have identified the outer and inner functions correctly
Check that you have found the derivative of each function using the correct rule (e.g. quotient rule or chain rule)
Check that you have multiplied the derivatives together correctly
Check that you have simplified the resulting expression correctly

Q: Can you give an example of how to use the chain rule in a programming language?

A: Yes, let's consider the example of using the chain rule in Python to find the derivative of a composite function. We can use the SymPy library to define the function and find its derivative using the chain rule. For example:

import sympy as sp

# Define the variables
t = sp.symbols('t')
a = sp.symbols('a')
k = sp.symbols('k')

# Define the function
p = 1 / (1 + a * sp.exp(-k * t))

# Find the derivative of the function using the chain rule
p_prime = sp.diff(p, t)

# Print the result
print(p_prime)

This code uses the SymPy library to define the function and find its derivative using the chain rule. The result is printed to the console.

Conclusion

In conclusion, the chain rule is a powerful tool in calculus that allows us to find the derivative of a composite function. By identifying the outer and inner functions, finding the derivative of each function using the correct rule, and multiplying the derivatives together, we can use the chain rule to find the derivative of a wide range of functions.