Differential Calculus and Multivariable Functions.
Understand the Problem
The question pertains to the topic of differential calculus and specifically focuses on multivariable functions, which involves understanding how to differentiate functions that depend on several variables.
Answer
Partial derivatives of $f(x, y) = x^2y + \sin(y)$ are $\frac{\partial f}{\partial x} = 2xy$ and $\frac{\partial f}{\partial y} = x^2 + \cos(y)$.
Answer for screen readers
The partial derivatives of the function $f(x, y) = x^2y + \sin(y)$ are:
$$ \frac{\partial f}{\partial x} = 2xy $$
and
$$ \frac{\partial f}{\partial y} = x^2 + \cos(y) $$
Steps to Solve
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Identify the function and variables Determine the function that you need to analyze and identify all its variables. For instance, let’s say we have a function $f(x, y)$.
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Determine the partial derivatives To find how $f$ changes with respect to one variable while keeping the others constant, calculate the partial derivatives. For example: $$ \frac{\partial f}{\partial x} $$ This represents the rate of change of $f$ concerning $x$ when $y$ is constant.
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Compute the partial derivatives Follow the rules of differentiation to compute the partial derivative. If $f(x, y) = x^2y + sin(y)$, then: $$ \frac{\partial f}{\partial x} = 2xy $$ and for $y$: $$ \frac{\partial f}{\partial y} = x^2 + cos(y) $$
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Evaluate at a specific point (if needed) If the problem requires finding the derivative at a specific point, substitute the given values into the computed derivatives. For example, evaluate $\frac{\partial f}{\partial x}$ at $(x, y) = (1, 0)$.
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Analyze the results Interpret the results of the partial derivatives. Understand what they indicate about the function's output as the input variables change.
The partial derivatives of the function $f(x, y) = x^2y + \sin(y)$ are:
$$ \frac{\partial f}{\partial x} = 2xy $$
and
$$ \frac{\partial f}{\partial y} = x^2 + \cos(y) $$
More Information
Partial derivatives are essential in multivariable calculus, as they allow you to understand how a function behaves when changes occur in different dimensions. This concept is applied in various fields such as physics, economics, and engineering.
Tips
- Confusing the concept of total derivatives with partial derivatives. It's important to remember that partial derivatives assume other variables are held constant.
- Forgetting to apply differentiation rules correctly, especially with respect to trigonometric functions or polynomials.
