Chapter 12: Problem 8
Find all first partial derivatives of each function. \(f(u, v)=e^{u v}\)
Short Answer
Expert verified
\(\frac{\partial f}{\partial u} = v e^{uv}, \frac{\partial f}{\partial v} = u e^{uv}\)
Step by step solution
01
Understand Partial Derivatives
To find the first partial derivatives of a function of two variables, you need to differentiate with respect to one variable while keeping the other constant. We are given the function \( f(u, v) = e^{uv} \). We will find \( \frac{\partial f}{\partial u} \) and \( \frac{\partial f}{\partial v} \).
02
Find the Partial Derivative with Respect to u
Apply the differentiation rules, treating \(v\) as a constant. Using chain rule, the derivative of \(e^{uv}\) with respect to \(u\) is \(v e^{uv}\) because the derivative of \(uv\) with respect to \(u\) is \(v\). Thus, \(\frac{\partial f}{\partial u} = v e^{uv}\).
03
Find the Partial Derivative with Respect to v
Now apply differentiation with respect to \(v\), treating \(u\) as a constant. Again using the chain rule, the derivative of \(e^{uv}\) with respect to \(v\) is \(u e^{uv}\) because the derivative of \(uv\) with respect to \(v\) is \(u\). Thus, \(\frac{\partial f}{\partial v} = u e^{uv}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multivariable Calculus
Multivariable calculus is an extension of calculus that deals with functions of multiple variables. While single-variable calculus deals with functions that have one independent variable, multivariable calculus considers functions that depend on two or more variables. This is crucial in many fields, such as physics, engineering, and economics, where systems are described by several interacting factors.
In multivariable functions, like the one given in the problem, we use partial derivatives to analyze how the function changes as each variable changes. A partial derivative is calculated by differentiating the function with respect to one variable, treating all other variables as constants. This approach helps us understand the role each variable plays in the system.
Consider a function \( f(u, v) = e^{uv} \). Here, \( u \) and \( v \) are the variables influencing the function. To extract how each variable individually affects the outcome of the function, we calculate the partial derivatives \( \frac{\partial f}{\partial u} \) and \( \frac{\partial f}{\partial v} \), which give us the rate of change in the function's value as \( u \) and \( v \) change, respectively.
In multivariable functions, like the one given in the problem, we use partial derivatives to analyze how the function changes as each variable changes. A partial derivative is calculated by differentiating the function with respect to one variable, treating all other variables as constants. This approach helps us understand the role each variable plays in the system.
Consider a function \( f(u, v) = e^{uv} \). Here, \( u \) and \( v \) are the variables influencing the function. To extract how each variable individually affects the outcome of the function, we calculate the partial derivatives \( \frac{\partial f}{\partial u} \) and \( \frac{\partial f}{\partial v} \), which give us the rate of change in the function's value as \( u \) and \( v \) change, respectively.
Chain Rule
The chain rule is a fundamental tool in calculus, allowing us to differentiate composite functions. When working with multivariable functions, the chain rule becomes a powerful technique to handle scenarios where functions are nested within other functions.
In the context of the exercise, our function \( f(u, v) = e^{uv} \) is an example of a composite function because it's a combination of the exponential function and the product \( uv \). To find partial derivatives like \( \frac{\partial f}{\partial u} \) and \( \frac{\partial f}{\partial v} \), the chain rule helps us "chain" the derivatives of these inner and outer functions together.
For instance, consider \( \frac{\partial f}{\partial u} \). Start by noticing that inside the exponent, \( uv \) is a product of two variables. When taking the partial derivative with respect to \( u \), \( v \) is treated as a constant. Hence, the derivative of \( uv \) with respect to \( u \) is \( v \). By the chain rule, we multiply this by the derivative of the outer function \( e^{uv} \), which remains \( e^{uv} \). Thus, \( \frac{\partial f}{\partial u} = v e^{uv} \).
This same process can be applied to \( \frac{\partial f}{\partial v} \), where \( u \) serves as the constant. The derivative of \( uv \) with respect to \( v \) is \( u \), leading to \( \frac{\partial f}{\partial v} = u e^{uv} \) by applying the chain rule again.
In the context of the exercise, our function \( f(u, v) = e^{uv} \) is an example of a composite function because it's a combination of the exponential function and the product \( uv \). To find partial derivatives like \( \frac{\partial f}{\partial u} \) and \( \frac{\partial f}{\partial v} \), the chain rule helps us "chain" the derivatives of these inner and outer functions together.
For instance, consider \( \frac{\partial f}{\partial u} \). Start by noticing that inside the exponent, \( uv \) is a product of two variables. When taking the partial derivative with respect to \( u \), \( v \) is treated as a constant. Hence, the derivative of \( uv \) with respect to \( u \) is \( v \). By the chain rule, we multiply this by the derivative of the outer function \( e^{uv} \), which remains \( e^{uv} \). Thus, \( \frac{\partial f}{\partial u} = v e^{uv} \).
This same process can be applied to \( \frac{\partial f}{\partial v} \), where \( u \) serves as the constant. The derivative of \( uv \) with respect to \( v \) is \( u \), leading to \( \frac{\partial f}{\partial v} = u e^{uv} \) by applying the chain rule again.
Exponential Functions
Exponential functions are a key concept in mathematics, particularly in calculus and modelling real-world phenomena. An exponential function, such as \( e^{x} \), involves a constant base raised to a variable power. This unique characteristic results in a rate of growth that becomes increasingly rapid.
The base \( e \) is known as the natural exponential base, approximately equal to 2.718. Its properties make it a frequent choice in calculus because \( e^{x} \)'s derivative is itself \( e^{x} \). This simplicity is handy when calculating derivatives and, especially, partial derivatives.
In the given function \( f(u, v) = e^{uv} \), \( e \) is raised to the power of \( uv \). This setup illustrates how exponential functions can also be part of more complex functions involving multiple variables. When calculating their derivatives, the base \( e \) ensures the exponential function behaves predictably, making it easier to focus on differentiating the exponent \( uv \) using rules like the chain rule.
Exponential functions' applications range widely from population growth models to financial calculations and beyond. Their distinctive ability to describe processes involving constant growth rates makes them an essential component of mathematical sciences.
The base \( e \) is known as the natural exponential base, approximately equal to 2.718. Its properties make it a frequent choice in calculus because \( e^{x} \)'s derivative is itself \( e^{x} \). This simplicity is handy when calculating derivatives and, especially, partial derivatives.
In the given function \( f(u, v) = e^{uv} \), \( e \) is raised to the power of \( uv \). This setup illustrates how exponential functions can also be part of more complex functions involving multiple variables. When calculating their derivatives, the base \( e \) ensures the exponential function behaves predictably, making it easier to focus on differentiating the exponent \( uv \) using rules like the chain rule.
Exponential functions' applications range widely from population growth models to financial calculations and beyond. Their distinctive ability to describe processes involving constant growth rates makes them an essential component of mathematical sciences.
