Problem 20
Question
2-20. Find the partial derivatives of \(f\) in terms of the derivatives of \(g\) and \(h\) if (a) \(f(x, y)=g(x) h(y) .\) (b) \(f(x, y)=g(x)^{h(y)}\). (c) \(f(x, y)=g(x)\). (d) \(f(x, y)=g(y)\) (e) \(f(x, y)=g(x+y)\)
Step-by-Step Solution
Verified Answer
For (a): \ \frac{\text{∂}f}{\text{∂}x} = h(y) g'(x), \ \frac{\text{∂}f}{\text{∂}y} = g(x) h'(y); For (b): \ \frac{\text{∂}f}{\text{∂}x} = h(y) g(x)^{h(y)-1} g'(x), \ \frac{\text{∂}f}{\text{∂}y} = g(x)^{h(y)} \text{ln}[g(x)] h'(y); For (c): \ \frac{\text{∂}f}{\text{∂}x} = g'(x), \ \frac{\text{∂}f}{\text{∂}y} = 0; For (d): \ \frac{\text{∂}f}{\text{∂}x} = 0, \ \frac{\text{∂}f}{\text{∂}y} = g'(y); For (e): \ \frac{\text{∂}f}{\text{∂}x} = g'(x + y), \ \frac{\text{∂}f}{\text{∂}y} = g'(x + y)
1Step 1 Title - Find the partial derivatives for Part (a)
Given the function: ewline (a) \(f(x, y) = g(x)h(y)\). ewline First, find the partial derivative with respect to x: ewline \ \frac{\text{∂}f}{\text{∂}x} = \frac{\text{∂}}{\text{∂}x} [g(x)h(y)] \ = h(y) \frac{\text{∂}}{\text{∂}x}[g(x)] \ = h(y) g'(x) ewline Now, find the partial derivative with respect to y: ewline \ \frac{\text{∂}f}{\text{∂}y} = \frac{\text{∂}}{\text{∂}y} [g(x)h(y)] \ = g(x) \frac{\text{∂}}{\text{∂}y}[h(y)] \ = g(x) h'(y)
2Step 2 Title - Find the partial derivatives for Part (b)
Given the function: ewline (b) \(f(x, y) = g(x)^{h(y)}\). ewline First, find the partial derivative with respect to x: ewline \ \frac{\text{∂}f}{\text{∂}x} = \frac{\text{∂}}{\text{∂}x} [g(x)^{h(y)}] \ = h(y) g(x)^{h(y)-1} \frac{\text{∂}}{\text{∂}x}[g(x)] \ = h(y) g(x)^{h(y)-1} g'(x) ewline Now, find the partial derivative with respect to y: ewline \ \frac{\text{∂}f}{\text{∂}y} = \frac{\text{∂}}{\text{∂}y} [g(x)^{h(y)}] \ = g(x)^{h(y)} \text{ln}[g(x)] \frac{\text{∂}}{\text{∂}y}[h(y)] \ = g(x)^{h(y)} \text{ln}[g(x)] h'(y)
3Step 3 Title - Find the partial derivatives for Part (c)
Given the function: ewline (c) \(f(x, y) = g(x)\). ewline Since the function depends only on x, the partial derivative with respect to x is: ewline \ \frac{\text{∂}f}{\text{∂}x} = g'(x) ewline The partial derivative with respect to y is zero: ewline \ \frac{\text{∂}f}{\text{∂}y} = 0
4Step 4 Title - Find the partial derivatives for Part (d)
Given the function: ewline (d) \(f(x, y) = g(y)\). ewline Since the function depends only on y, the partial derivative with respect to y is: ewline \ \frac{\text{∂}f}{\text{∂}y} = g'(y) ewline The partial derivative with respect to x is zero: ewline \ \frac{\text{∂}f}{\text{∂}x} = 0
5Step 5 Title - Find the partial derivatives for Part (e)
Given the function: ewline (e) \(f(x, y) = g(x + y)\). ewline Find the partial derivative with respect to x: ewline \ \frac{\text{∂}f}{\text{∂}x} = \frac{\text{∂}}{\text{∂}x}[g(x + y)] \= g'(x + y) \frac{\text{∂}}{\text{∂}x}[x + y] \ = g'(x + y)(1) \ = g'(x + y) ewline Now, find the partial derivative with respect to y: ewline \ \frac{\text{∂}f}{\text{∂}y} = \frac{\text{∂}}{\text{∂}y}[g(x + y)] \= g'(x + y) \frac{\text{∂}}{\text{∂}y}[x + y] \ = g'(x + y)(1) \ = g'(x + y)
Key Concepts
calculusmultivariable functionsdifferentiation
calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. One of its key components is differentiation, which involves finding the derivative of a function. Derivatives represent how a function changes as its input changes, providing vital information in various fields such as physics, engineering, and economics.
Calculus explores both single-variable and multivariable functions. In single-variable calculus, you're often finding the rate of change of a function with respect to one variable. When dealing with functions of several variables, you encounter partial derivatives, which measure how a function changes with respect to one variable while keeping the others constant.
For example, if you have a function like \( f(x,y) = g(x)h(y) \), you can take the partial derivative of \( f \) with respect to \( x \) as \( \frac{\partial f}{\partial x} = h(y) g'(x) \), while the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = g(x) h'(y) \). This approach helps in understanding how each variable individually influences the function.
Calculus explores both single-variable and multivariable functions. In single-variable calculus, you're often finding the rate of change of a function with respect to one variable. When dealing with functions of several variables, you encounter partial derivatives, which measure how a function changes with respect to one variable while keeping the others constant.
For example, if you have a function like \( f(x,y) = g(x)h(y) \), you can take the partial derivative of \( f \) with respect to \( x \) as \( \frac{\partial f}{\partial x} = h(y) g'(x) \), while the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = g(x) h'(y) \). This approach helps in understanding how each variable individually influences the function.
multivariable functions
Multivariable functions are functions that depend on two or more variables. They extend the concepts from single-variable calculus to functions involving several variables. This is particularly useful in real-world scenarios where outcomes depend on multiple factors.
When dealing with multivariable functions, it's crucial to understand the concept of partial derivatives. Partial derivatives give us the rate of change of the function with respect to one variable, holding the others constant. For instance, consider the function \( f(x, y) = g(x)^{h(y)} \). The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = h(y) g(x)^{h(y)-1} g'(x) \), while the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = g(x)^{h(y)} \ln(g(x)) h'(y) \).
Finding partial derivatives is similar to regular differentiation but focuses on one variable at a time. This technique is invaluable for analyzing how each input variable affects the overall function independently.
When dealing with multivariable functions, it's crucial to understand the concept of partial derivatives. Partial derivatives give us the rate of change of the function with respect to one variable, holding the others constant. For instance, consider the function \( f(x, y) = g(x)^{h(y)} \). The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = h(y) g(x)^{h(y)-1} g'(x) \), while the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = g(x)^{h(y)} \ln(g(x)) h'(y) \).
Finding partial derivatives is similar to regular differentiation but focuses on one variable at a time. This technique is invaluable for analyzing how each input variable affects the overall function independently.
differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function's value changes as its input changes. In calculus, differentiation forms the foundation for understanding how functions behave.
For functions that depend on multiple variables, we use partial derivatives to understand how the function changes with respect to each variable independently. For example, if \( f(x, y) = g(x + y) \), the partial derivatives are given by \( \frac{\partial f}{\partial x} = g'(x + y) \) and \( \frac{\partial f}{\partial y} = g'(x + y) \). This shows that the function's rate of change with respect to \( x \) and \( y \) is the same if their sum remains the same.
Differentiation in multivariable functions offers insights into how changes in one variable affect the entire system, making it an essential tool in mathematical modeling and analysis.
For functions that depend on multiple variables, we use partial derivatives to understand how the function changes with respect to each variable independently. For example, if \( f(x, y) = g(x + y) \), the partial derivatives are given by \( \frac{\partial f}{\partial x} = g'(x + y) \) and \( \frac{\partial f}{\partial y} = g'(x + y) \). This shows that the function's rate of change with respect to \( x \) and \( y \) is the same if their sum remains the same.
- Step-by-Step Differentiation:
- Identify the function and the variable you want to differentiate with respect to.
- Apply the rules of differentiation (such as the product rule, quotient rule, or chain rule).
- Simplify the resulting expression to find the partial derivative.
Differentiation in multivariable functions offers insights into how changes in one variable affect the entire system, making it an essential tool in mathematical modeling and analysis.
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